Homogenization of Griffith’s Criterion for Brittle Laminates

Abstract

We consider a periodic, linear elastic laminate with a brittle crack, evolving along a prescribed path according to Griffith’s criterion. We study the homogenized limit of this evolution, as the size of the layers vanishes. The limit evolution is governed again by Griffith’s criterion, in terms of the energy release (of the homogenized elastic energy) and an effective toughness, which, in general, differs from the $\hbox {weak}^*$ limit of the periodic toughness. We provide a variational characterization of the effective toughness and, by the energy identity, we link the toughening effect (in the limit) to the micro-instabilities of the evolution (in the periodic laminate). Finally, we provide a couple of explicit calculations of the effective toughness in the anti-plane setting, showing, in particular, an example of toughening by elastic contrast.

1 Introduction

From the mathematical point of view, the homogenization of evolution equations is not a simple development of the (well established) homogenization theory for elliptic equations. A remarkable example was presented by Tartar in [23]. Consider a Cauchy problem of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{u}_\varepsilon ( x , t ) + a_\varepsilon (x) u _\varepsilon ( x, t) = f( x, t ) \\ u_\varepsilon ( x, 0 ) = u^0 (x) . \end{array}\right. } \end{aligned}$$

If $a_\varepsilon {\mathop {\rightharpoonup }\limits ^{*}}a_0$ (which is the case in homogenization), then, up to subsequences, $u_\varepsilon \rightharpoonup u_0$, where $u_0$ solves an evolution equation of the form

$$\begin{aligned} \dot{u}_0 ( x , t ) + a_0 (x) u_0 ( x, t) + \int _0^t K ( x, t- \tau ) u (\tau ) \, \textrm{d}\tau = f( x, t ) . \end{aligned}$$

Therefore, a memory effect (in time) appears in the limit, as a consequence of the heterogeneity (in space) of the coefficients. This example was studied also in [15, Section 3.5.2] from a different perspective: setting the problem in spaces of measures, Mielke showed that different limit evolution may occur, as the system is endowed with different energy-dissipation structures. For instance, if $a_\varepsilon (x) = a ( x / \varepsilon )$ and $f=0$, then suitable choices of energy and dissipation lead to the following limit equations:

$$\begin{aligned} \dot{u}_0 ( x , t ) + ( \min a ) \, u_0 ( x, t) = 0 , \quad \dot{u}_0 ( x , t ) + ( \max a) \, u_0 ( x, t) = 0 \end{aligned}$$

(clearly, different subsequences are extracted from $u_\varepsilon$). In [15] this example is presented in the context of evolutionary $\Gamma$-convergence, which gives a notion of convergence for a class of doubly non-linear parabolic evolutions of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{v} \mathcal {R}_\varepsilon ( u_\varepsilon (x,t) , \dot{u}_\varepsilon (x,t) ) = - \nabla \mathcal {E}_\varepsilon ( u_\varepsilon (x,t) ) , \\ u_\varepsilon ( 0, x) = u^\varepsilon (x) , \end{array}\right. } \end{aligned}$$

usually when $\mathcal {E}_\varepsilon$ $\Gamma$-converge to $\mathcal {E}_0$ (for the general theory of $\Gamma$-convergence we refer to the book of Dal Maso [10]). When the dissipation is simply of the form $\mathcal {R}_\varepsilon ( u , v ) = \tfrac{1}{2}\Vert v \Vert ^2$, the abstract results of Sandier and Serfaty [22] provide sufficient conditions which guarantee the (weak) convergence of $u_\varepsilon$ to the solution $u_0$ of the parabolic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{u}_0 (x,t) = - \nabla \mathcal {E}_0 ( u_0 (x,t) ) , \\ u_0 ( 0, x) = u^0 (x) . \end{array}\right. } \end{aligned}$$

For instance, under suitable regularity assumptions, convergence holds if the initial data is well-prepared, that is, if $u^\varepsilon \rightharpoonup u^0$ and $\mathcal {E}_\varepsilon (u^\varepsilon ) \rightarrow \mathcal {E}_0 ( u^0)$, and if

$$\begin{aligned} \Vert \nabla \mathcal {E}_0 \Vert ( u ) \leqq \Gamma \text {-}\liminf _{\varepsilon \rightarrow 0} \Vert \nabla \mathcal {E}_\varepsilon \Vert ( u) . \end{aligned}$$

In the rate-independent setting, similar conditions are sufficient also for the convergence of balanced (or vanishing) viscosity evolutions; see for example [19]. The conditions in [22] are actually far more general, and include the case of $\varepsilon$-depending norms

$$\begin{aligned} \Vert \nabla \mathcal {E}_0 \Vert _0 ( u ) \leqq \Gamma \text {-}\liminf _{\varepsilon \rightarrow 0} \Vert \nabla \mathcal {E}_\varepsilon \Vert _\varepsilon ( u) , \end{aligned}$$

(1)

which will be relevant also in our context. Finally, consider a family of minimizing movement (or implicit Euler) schemes of the form

$$\begin{aligned} u_\varepsilon ( (k+1) \tau _\varepsilon ) \in \textrm{argmin} \left\{ \mathcal {E}_\varepsilon (u) + \tfrac{1}{2} \tau _\varepsilon ^{-1} \Vert u - u_\varepsilon ( k \tau _\varepsilon ) \Vert ^2 \right\} , \end{aligned}$$

where $\tau _\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$. For when the energy $\mathcal {E}_\varepsilon$ is independent of $\varepsilon$, it is well known that (the interpolation of) the above time-discrete evolutions converge, as $\varepsilon \rightarrow 0$, to the gradient flow of $\mathcal {E}_\varepsilon$. However, by virtue of the examples provided in Braides [8], the limit evolution will, in general, depend on the relationship between $\varepsilon$ (which parametrizes the family of energies) and the time step $\tau _\varepsilon$.

From the mechanical point of view our work is motivated by experimental and numerical results on the effective toughness of brittle composites (see for instance [13, 24] and the references therein). In general, when dealing with a periodically arranged composite, with a very fine microstructure, it is natural to introduce also an homogenized material reflecting “in average” the mechanical behavior of the composite. For instance, the equilibrium configuration (in terms of displacement, stress and energy) of a linear elastic composite can be approximated by the equilibrium configuration of a homogeneous linear elastic material, obtained with “suitable averages” of the stiffness tensors of the underlying constituents.

For quasi-static fracture propagation, the mechanical behavior is instead described by the crack set, at time t, together with the equilibrium configuration of the body, given the crack at time t; furthermore, the propagation is governed by a rate-independent evolution law for the crack (Griffith’s criterion), together with momentum balance (an elliptic pde). For the latter, (static) homogenization is suitable, and several mathematical approaches are available in the literature; see for example [5, 9, 17] and the references therein. For the former, that is, Griffith’s criterion, there is a need instead to develop some kind of “evolutionary homogenization” or, better, an “evolutionary convergence” for rate-independent systems (specified hereafter). In practice, see for instance [13, 24], the goal would be to find an effective “averaged” toughness in such a way that the quasi-static evolution of a crack in the brittle composite is approximated by the quasi-static evolution of a crack in a brittle homogeneous material characterized by the (static) homogenization of the stiffness tensors together with the effective toughness. In §2.8, we provide an example of a laminate with explicit computations of effective toughness and homogenized stiffness. This example highlights some peculiar features: the effective toughness depends on the volume fraction of the layers, on their toughness, and also on the coefficients of their stiffness matrices, furthermore, it can be larger than the toughness of the single brittle layers, and larger than the $\hbox {weak}^*$ limit of the periodic toughness. These properties are qualitatively consistent with the numerical and experimental results of Hossain et al. [13], where toughening (that is, the macroscopic increase in toughness) occurs as a result of the microscopic elastic and toughness heterogeneities. In mechanics, most of the interest is indeed driven by the toughening and the strengthening effects of composites [6]. In general, toughening is also due to several microscopic “geometrical features” of the crack, such as micro-cracking, branching, debonding and tortuosity; a mathematical result on toughening by the homogenization of highly oscillating crack paths is given in [4], where Barchiesi employs a modified $\Gamma$-convergence framework in order to take into account the irreversibility constraint. In the context of Coulomb friction, a characterization of the effective friction coefficient (depending on both elastic and friction coefficients of the layers) is provided in [3].

Let us turn to our results. We consider a brittle crack which propagates horizontally in a linear elastic laminate composed by n periodic layers; each layer is made of a layer of material A and B with volume ratio $\lambda$ and $1-\lambda$, respectively. We distinguish between vertical [13] and horizontal layers: in the former setting, the crack crosses the layers, while in the latter it lays in the interface between two layers. We do not consider a “surfing boundary condition” [13], but a Dirichlet boundary condition $u = f(t) g$ for the displacement u on a portion $\partial _D \Omega$ of the boundary. For technical reasons, and making reference to the experimental and numerical results of [13], we confine a priori the geometry of the crack to a straight line, even if, in general, the crack may deflect at interfaces; see the well-known [12], which leads to (macroscopic) anisotropic toughness in layered materials [7].

We begin with the more delicate case, which is that of vertical layers. We denote the toughness by $G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ and the energy release by $G_n (t , l )$, where t is time and l indicates the position of the crack tip. Note that $G^{\hspace{-2.5pt}{ \textit{c}}}_n$ depends on l and that it is periodic. On the contrary, $G_n (t, \cdot )$ is not periodic and, to the best of our knowledge, it is not known if it is defined at the interfaces between the two materials; for this reason, we actually employ the right lower semi-continuous envelope, which is indeed the natural choice for Griffith’s criterion (see Remark 2). By linearity, the dependence on time is much easier, since we can write $G_n (t, l) = f^2(t) \hspace{0.5pt}\mathcal {G}_n (l)$, where $\hspace{0.5pt}\mathcal {G}_n$ is computed with boundary condition $u = g$, that is, for $f(t) =1$. Denoting by $\ell _n(t)$ the position of the tip at time t, the quasi-static propagation is governed by Griffith’s criterion:

• (i) $G_n (t, \ell _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n(t))$,

• (ii) if $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t))$ then $\dot{\ell }^+_n (t) =0$ (that is, the right time derivative vanishes).

It is well known, see for example [18, 21], that, in general, there are different solutions satisfying Griffith’s criterion, and that these solutions may be discontinuous (in time). Discontinuities are basically due to the fact that Griffith’s criterion is rate-independent (that is, invariant under time reparametrization). From the mathematical point of view, the behavior in the discontinuities characterizes the different notions of evolution; see [18]. Here, we denote by $J(\ell _n)$ the set times where $\ell _n$ jumps, and following [21], we require, besides (i) and (ii), that

• (iii) if $t \in J ( \ell _n)$ then $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [\ell ^-_n(t) , \ell ^+_n(t) )$.

This condition means that the transition between the equilibria $\ell ^-_n(t)$ and $\ell ^+_n(t)$ is unstable (or catastrophic). Notice that, for $l = \ell ^-_n(t)$, the above conditions, put together, imply that $G_n (t, \ell ^-_n(t))=G^c_n ( \ell _n^- (t))$, therefore both stable and unstable propagation occurs once the critical value is attained. Evolutions satisfying (i), (ii), and (iii) can be obtained also by a vanishing viscosity approach [14], taking the limit of continuous solutions of rate-dependent models. As observed in the numerical simulations [13], discontinuities often occur at those interfaces where the crack passes from the material with higher to that with lower toughness. Therefore, in our evolution $\ell _n$, we should expect many of these “microscopic jumps” in the order n of the number of layers; these discontinuities, which “disappear” in the homogenized limit, will produce the toughening effect.

At this point, let us describe the homogenized limit. Denote by $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l)$ the energy release of the homogenized elastic energy. By Helly’s theorem we know that $\ell _n \rightarrow \ell$ pointwise (upon extracting a subsequence). The goal is to define an effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ in such a way that the limit $\ell$ satisfies Griffith’s criterion:

• (i) $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \ell (t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell (t))$,

• (ii) if $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, \ell (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell (t))$ then $\dot{\ell }^+ (t) =0$ (that is, the right time derivative vanishes),

• (iii) if $t \in J ( \ell )$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ for every $l \in [\ell ^- (t) , \ell ^+ (t) )$.

Notice that, in general, $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \cdot )$ is neither the pointwise nor the $\Gamma$-limit of $G_n (t,\cdot )$ (an explicit example is given in Section 2.8). However, a variational convergence in the spirit of (1) holds. Indeed, writing again that $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , l ) = f^2(t) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$, the effective toughness is defined in such a way that

$$\begin{aligned} \displaystyle \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) } = \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l) }{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) } \,. \end{aligned}$$

(2)

First of all, note that, in general, $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ may depend on l and, through the energy release, on several other parameters of the problem, in particular, it does not depend only on the toughness of material A and B; in mechanical terms, it is an R-curve (see for example [2]). At the current stage, it is hard to guess if $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ should always be a constant. In our examples it is so, on the contrary, in the experimental measures of [6], the effective toughness is surely not constant. What is clear is the fact that the effective toughness depends on the elastic contrast and it can be higher than the toughness of the underlying materials. Indeed, in the explicit calculations of Section 2.8, we show that (under suitable assumptions)

$$\begin{aligned} \displaystyle G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}= \lambda \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\, \frac{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} \right\} + ( 1 - \lambda ) \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\frac{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} \, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\right\} , \end{aligned}$$

where $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$ are the toughness, while $\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}$ and $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ are the components of the stiffness matrix in the anti-plane setting. Moreover, independently of this example, if $\hspace{0.5pt}\mathcal {G}_n$ converge uniformly to $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, then $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}= \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$.

Finally, let us compare (2) with (1). In the context of fracture, the energy release $\hspace{0.5pt}\mathcal {G}_n$ plays the role of the slope $\Vert \nabla \mathcal {E}_n \Vert$; the ratio $\hspace{0.5pt}\mathcal {G}_n / G^{\hspace{-2.5pt}{ \textit{c}}}_n$ is instead some sort of energy release with respect to the dissipation metric induced by the toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_n$, and plays the role of $\Vert \nabla \mathcal {E}_n \Vert _n$; in these terms, the effective toughness plays the role of the norm $\Vert \cdot \Vert _0$ in (1) or, more precisely, of the smallest norm which makes (1) true, and for this reason in the definition of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ an equality is needed.

In the case of horizontal layers with an interface crack the picture is much simpler. Indeed, the toughness $G^{\hspace{-2.5pt}{ \textit{c}}}$ is constant and independent of n, and moreover the energy release $\hspace{0.5pt}\mathcal {G}_n$ is well defined and converges locally uniformly to $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$; this is enough to prove that the evolutions $\ell _n$ converge to an evolution $\ell$ which satisfies Griffith’s criterion with energy release $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}= G^{\hspace{-2.5pt}{ \textit{c}}}$.

2 Vertical Layers with Horizontal Crack

Let $\Omega = (0,L) \times (-H,H)$ be the (uncracked) reference configuration. For $l \in (0,L]$ let $K_l = [0, l] \times \{0\}$ be the crack; in the presence of a crack $K_l$ the reference configuration is then $\Omega {\setminus } K_l$. We denote by $\partial _D \Omega$, independently of l, the union of the sets $\{ 0 \} \times ( -H,0) \cup (0,H)$ and $\{ L \} \times (-H,H)$. Let $g \in H^{1/2} ( \partial _D \Omega )$. To fix the ideas,

$$\begin{aligned} g (x) = {\left\{ \begin{array}{ll} -1 & x \in \{ 0 \} \times (-H,0), \\ 1 & x \in \{ 0 \} \times (0,H) , \\ 0 & x \in \{ L \} \times (-H,H) \end{array}\right. } \end{aligned}$$

is an admissible function (see also Fig. 1). By abuse of notation we still denote by g a lifting of the boundary datum. Given $l \in (0,L]$, the spaces of admissible displacements and admissible variations are, respectively,

$$\begin{aligned} \mathcal {U}_{{\hspace{0.5pt}l}} = \{ u \in H^1(\Omega {\setminus } K_l) : u = g \text { in } \partial _D \Omega \} , \quad \mathcal {V}_{{\hspace{0.5pt}l}} = \{ v \in H^1(\Omega {\setminus } K_l) : v = 0 \text { in }\partial _D \Omega \} . \end{aligned}$$

Note that $\mathcal {U}_l \ne \emptyset$, since $g \in H^{1/2} ( \{ L \} \times (-H,H) )$. Clearly, if $l_1 < l_2$, then $\mathcal {U}_{{\hspace{0.5pt}l}_1} \subset \mathcal {U}_{{\hspace{0.5pt}l}_2}$, and thus $\mathcal {U}_{{\hspace{0.5pt}l}} \subset \mathcal {U}_L \subset H^1(\Omega {\setminus } K_L)$ for every $l \in (0,L]$. All the spaces $\mathcal {V}_{{\hspace{0.5pt}l}}$ and the sets $\mathcal {U}_{{\hspace{0.5pt}l}}$ are endowed with the $H^1 ( \Omega {\setminus } K_L)$-norm. Splitting $\Omega {\setminus } K_L$ into the disjoint union of $\Omega ^+ = (0,L) \times (0,H)$ and $\Omega ^- = (0,L) \times (-H,0)$, a uniform Poincaré inequality holds: there exists $C_P>0$ such that, for every $l \in (0,L]$ and $v \in \mathcal {V}_l$,

$$\begin{aligned} \int _{ \Omega {\setminus } K_l} | v |^2 \, \textrm{d}x \leqq C_P \int _{\Omega {\setminus } K_l} | \nabla v |^2 \, \textrm{d}x . \end{aligned}$$

Fig. 1

An example of vertical layers and horizontal crack (along the mid line)

The conditions given above are independent of the periodic structure and hold throughout the paper, for both abstract results and explicit computations. Actually, some of the results hold under more general conditions, depending on the specific setting. For instance, the abstract results hold also in the plane-strain setting, see Section 2.6.1, however, in this context we do not have any explicit example; for this reason and for the sake of simplicity, we consider the antiplane setting presented above.

2.1 Energy and Energy Release

In this section we assume that $\Omega$ is a composite, made of $n \in \mathbb {N} {\setminus } \{ 0\}$ periodic vertical layers of thickness $l_n = L/n$; each layer itself is composed of a vertical layer of material A with thickness $\lambda l_n$, for $\lambda \in (0,1)$, and a vertical layer of material B with thickness $(1-\lambda ) l_n$. More precisely, let $l_{n,k} = k\, l_n$ for $k=0,\ldots ,n$ and $l_{n,k+\lambda } = {\hspace{0.5pt}l}_{n,k} + \lambda l_n = (k + \lambda ) l_n$ for $k=0,\ldots ,n-1$; the layers of materials A are of the form $(l_{n,k}, l_{n,k+\lambda }) \times (-H,H)$ while those of material B are of the form $( l_{n,k+\lambda } , l_{n,k+1}) \times (-H,H)$. For later convenience, we also introduce the notation $\Lambda _{n} = \{ l \in [0,L] : l = l_{n,k} \text { or } l = l_{n,k+\lambda } \}$ so that the interfaces between material A and B (in the sound material) are of the form $\{ l \} \times (-H, H)$ for $l \in \Lambda _n {\setminus } \{ 0, L \}$. We will also employ the notation $\Omega _{n,k} = ( l_{n,k} , l_{n,k+1}) \times (-H,H)$ and

$$\begin{aligned} \Omega _{n,\hspace{1pt}{\hspace{-4pt} A}} = \bigcup _{k=0}^{n-1} \ ( l_{n,k} , l_{n,k+\lambda }) \times (-H,H) , \quad \Omega _{n,\hspace{1pt}{\hspace{-3.5pt} B}} = \bigcup _{k=0}^{n-1} \ ( l_{n,k +\lambda } , l_{n,k+1}) \times (-H,H) . \end{aligned}$$

We define in a similar way the crack sets $K_{n,k} = ( l_{n,k} , l_{n,k+1} ) \times \{ 0 \}$, and then $K_{n,\hspace{1pt}{\hspace{-4pt} A}}$ and $K_{n,\hspace{1pt}{\hspace{-3.5pt} B}}$.

We assume that A and B are elastic brittle materials. We denote by $\varvec{C}_{\hspace{-4pt} A}$ the stiffness matrix of material A, of the form

$$\begin{aligned} \varvec{C}_{\hspace{-4pt} A}= \left( \begin{matrix} \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} \end{matrix} \right) , \quad \mu _{{\hspace{-4pt} A},i} >0\text { for }i=1,2, \end{aligned}$$

and we denote by $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}>0$ the toughness of A; we employ a similar notation for material B. Let $\, \varvec{C}_n : \Omega {\setminus } ( \Lambda _n \times (-H,H)) \rightarrow \{ \varvec{C}_{\hspace{-4pt} A}, \varvec{C}_{\hspace{-3.5pt} B}\}$ be the periodic stiffness matrix and let $G^{\hspace{-2.5pt}{ \textit{c}}}_n : [0,L] {\setminus } \Lambda _n \rightarrow \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$ be the periodic toughness.

The linear elastic energy $\mathcal {W}_{l,n}: \mathcal {U}_l \rightarrow \mathbb {R}$ is given by

$$\begin{aligned} \mathcal {W}_{l,n} (u) = \tfrac{1}{2} \int _{\Omega {\setminus } K_l} \nabla u \, \varvec{C}_n \nabla u^T \, \textrm{d}x . \end{aligned}$$

Denote by $u_{l,n} \in \mathcal {U}_{{\hspace{0.5pt}l}}$ the (unique) minimizer of the energy $\mathcal {W}_{l,n}$; clearly $u_{l,n}$ is also the (unique) solution of the variational problem

$$\begin{aligned} \int _{\Omega {\setminus } K_l} \nabla u \, \varvec{C}_n \nabla v^T \, \textrm{d}x = 0 \quad \text {for every }v \in \mathcal {V}_l. \end{aligned}$$

Note that the above bi-linear form is coercive and continuous, uniformly with respect to $l \in (0,L]$ and $n \in \mathbb {N}$. In the sequel we will often employ the condensed (or reduced) elastic energy $\mathcal {E}_n : (0,L] \rightarrow \mathbb {R}$ given by

$$\begin{aligned} \mathcal {E}_n ( l ) = \mathcal {W}_{l,n} ( u_{l,n} ) = \min \{ \mathcal {W}_{l,n} ( u) : u \in \mathcal {U}_{{\hspace{0.5pt}l}} \} . \end{aligned}$$

The following lemma contains the fundamental properties of this energy (for n fixed):

Lemma 1

The energy $\mathcal {E}_n$ is decreasing, continuous, and of class $C^1 ( (0,L) {\setminus } \Lambda _n )$. In particular, if $l \in (l_{n,k} , l_{n,k+\lambda })$ the derivative takes the form

$$\begin{aligned} \mathcal {E}'_n ( l) = \int _{\Omega {\setminus } K_l} \nabla u_{l,n} \varvec{C}_n \varvec{E} \nabla u_{l,n}^T \, \phi ' \textrm{d}x \quad \text {where} \quad \varvec{E} = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) , \end{aligned}$$

(3)

while $\phi \in W^{1,\infty } (0,L)$, $\textrm{supp} ( \phi ) \subset ( l_{n,k} , l_{n,k+\lambda } )$, $\phi (l) = 1$ and, by abuse of notation, $\phi ' = \partial _{l} \phi (x_1)$. A representation like (3) holds also for $l \in (l_{n,k+\lambda }, l_{n,k+1})$.

Remark 1

Note that $\mathcal {E}_n$ is not periodic, unless it is constant. Note also that the support of the auxiliary function $\phi$ shrinks when $n \rightarrow \infty$.

Proof

We provide a short proof, following [20]. For $l_1 < l_2$ we have $\mathcal {U}_{{\hspace{0.5pt}l}_1} \! \subset \mathcal {U}_{{\hspace{0.5pt}l}_2}$ and thus $\mathcal {E}_n ( l_1) \geqq \mathcal {E}_{n} ( l_2)$. If $l_m \rightarrow l$, then $\mathcal {E}_n (l_m) \rightarrow \mathcal {E}_n ( l)$ and $u_{l_m,n} \rightarrow u_{l,n}$ strongly in $H^1 (\Omega {\setminus } K_L)$.

Let us compute the derivative of the energy $\mathcal {E}_n (l)$ for $l \not \in \Lambda _n$. For $h \ll 1$ let $\Psi _h ( x) = (x_1 + h \phi (x_1), x_2)$ be a diffeomorphism of $\Omega {\setminus } K_{l}$ onto $\Omega {\setminus } K_{l+h}$. Note that $\Psi _h (x) = x$ out of the layer $(l_{n,k} , l_{n,k+\lambda } ) \times (-H,H)$. Using the change of variable $w = u \circ \Psi _h$ the energy $\mathcal {W}_{l+h,n} : \mathcal {U}_{\,l+h} \rightarrow \mathbb {R}$ reads as

$$\begin{aligned} \mathcal {W}_{l+h,n} ( u ) = \overline{\mathcal {W}}_{l,h,n} (w) = \tfrac{1}{2} \int _{\Omega {\setminus } K_l} \nabla w \, \varvec{C}_n ( \varvec{I} + h \varvec{M}_{\!h} \phi ' ) \nabla w^T \, \textrm{d}x, \end{aligned}$$

where

$$\begin{aligned} \varvec{M}_{\!h} = \left( \begin{array}{cc} - ( 1 + h \phi ')^{-1} & 0 \\ 0 & 1 \end{array} \right) . \end{aligned}$$

If $u_{l+h,n} \in \textrm{argmin} \,\{ \mathcal {W}_{l+h,n} (u) : u \in \mathcal {U}_{\,l+h} \}$, then $w_{l,h,n} = (u_{l+h,n} \circ \Psi _h) \in \textrm{argmin} \,\{ \overline{\mathcal {W}}_{l,h,n} (w) : w \in \mathcal {U}_{\,l} \}$ and $\mathcal {E}_n (l+h) = \overline{\mathcal {W}}_{l,h,n} ( w_{l,h,n})$. Hence,

$$\begin{aligned} \mathcal {E}'_n (l) = \lim _{h \rightarrow 0} \frac{\mathcal {E}_n (l+h) - \mathcal {E}_n (l)}{h} = \lim _{h \rightarrow 0} \frac{\overline{\mathcal {W}}_{l,h,n} (w_{l,h,n}) - \mathcal {W}_{l,n} (u_{l,n})}{h} . \end{aligned}$$

Writing the variational formulations for $w_{l,h,n}$ and $u_{l,n}$, we have

$$\begin{aligned} \int _{\Omega {\setminus } K_l} \nabla w_{l,h,n} \, \varvec{C}_n ( \varvec{I} + h \varvec{M}_{\!h} \phi ' ) \nabla v^T \, \textrm{d}x = \int _{\Omega {\setminus } K_l} \nabla u_{l,n} \, \varvec{C}_n \nabla v^T \, \textrm{d}x = 0 \quad \text {for every }v \in \mathcal {V}_{{\hspace{0.5pt}l}}. \end{aligned}$$

Then, being that $(w_{l,h,n} - u_{l,n}) \in \mathcal {V}_l$, we can re-write the energies as

$$\begin{aligned} \overline{\mathcal {W}}_{l,h,n} (w_{l,h,n})&= \tfrac{1}{2} \int _{\Omega {\setminus } K_l} \nabla w_{l,h,n} \, \varvec{C}_n ( \varvec{I} + h \varvec{M}_{\!h} \phi ' ) \nabla u_{l,n}^T \, \textrm{d}x ,\\ \mathcal {W}_{l,n} (u_{l,n})&= \tfrac{1}{2} \int _{\Omega {\setminus } K_l} \nabla w_{l,h,n} \, \varvec{C}_n \nabla u_{l,n}^T \, \textrm{d}x . \end{aligned}$$

Hence,

$$\begin{aligned} \frac{ \overline{\mathcal {W}}_{l,h,n} (w_{l,h,n}) - \mathcal {W}_{l,n} (u_{l,n})}{h}= \tfrac{1}{2} \int _{\Omega {\setminus } K_l} \nabla w_{l,h,n} \, \varvec{C}_n \varvec{M}_{\!h} \nabla u_{l,n}^T \, \phi ' \, \textrm{d}x . \end{aligned}$$

As $h \rightarrow 0$, $\varvec{M}_{\!h} \rightarrow 2 \varvec{E}$ in $L^\infty (\Omega ; \mathbb {R}^{2 \times 2} )$ and $w_{l,h,n} \rightharpoonup u_{l,n}$ in $H^1( \Omega {\setminus } K_l)$, therefore we obtain (3).

If $l_m \rightarrow l$ and $l_{n,k}< l < l_{n,k+\lambda }$ we can choose $\phi$ to be independent of m (sufficiently large). Moreover, $\nabla u_{l_m,n} \rightarrow \nabla u_{l,n}$ in $L^2(\Omega {\setminus } K_L , \mathbb {R}^2)$. Therefore, by the representation (3) we get $\mathcal {E}'_n (l_m) \rightarrow \mathcal {E}'_n (l)$. $\square$

By virtue of the above Proposition, for every $l \not \in \Lambda _n$, we can define the energy release

$$\begin{aligned} \mathcal {G}_n (l) = - \partial ^+_l \mathcal {E}_n (l) = - \partial _l \mathcal {E}_n (l) . \end{aligned}$$

(4)

In the sequel it will be convenient to extend the toughness and the energy release to $\Lambda _n$ (even if in general the energy is not differentiable in $\Lambda _n$). We define

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_n ( l ) = \liminf _{s \rightarrow l^+} \hspace{0.5pt}\mathcal {G}_n ( s) \text { if }l \ne L \quad \text { and } \quad \hspace{0.5pt}\mathcal {G}_n ( L) = 0 . \end{aligned}$$

(5)

With this definition the function $\hspace{0.5pt}\mathcal {G}_n : [0,L] \rightarrow [0,+\infty ]$ is right lower semi-continuous. In Remark 2 we will see that for $l \ne L$ this extension is actually the only one compatible with Griffith’s criterion; for $l=L$ it is instead suggested by technical convenience (strictly speaking, the right derivative of the reduced energies $\mathcal {E}_n$ do not make sense for $l=L$). Note that $\hspace{0.5pt}\mathcal {G}_n$ may possibly take the value $+\infty$ in $\Lambda _n$. In a similar manner, we define

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) = \lim _{s \rightarrow l^+} G^{\hspace{-2.5pt}{ \textit{c}}}_n (s)\text { if }l \ne L \quad \text { and } \quad G^{\hspace{-2.5pt}{ \textit{c}}}_n (L) = G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}. \end{aligned}$$

(6)

Clearly, $G_n^c : [0,L] \rightarrow \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$ is right continuous.

2.2 Quasi-Static Evolution by Griffith’s Criterion

In this subsection we provide a notion of quasi-static propagation, following [14, 18, 21]. We consider a time depending boundary datum of the form f(t)g where $f \in C^1([0,T])$ with $f(0)=0$ and $f' > 0$ (non-monotone boundary conditions are considered in Section 4). Let $L_0 \in (0,L)$ be the initial crack. For $l \in [L_0,L]$ the space of admissible displacement is

$$\begin{aligned} U_{t,l} = \{ u \in H^1(\Omega {\setminus } K_l) : u = f(t) g \text { in } \partial _D \Omega \} = f(t) \, \mathcal {U}_{{\hspace{0.5pt}l}} . \end{aligned}$$

We denote by $E_n$ and $G_n$ the corresponding condensed elastic energy and energy release. For $t=0$ we simply have that $E_n (0, \cdot ) = G_n (0, \cdot ) = 0$, since $u=0$ on $\partial _D \Omega$. If $t>0$, by linearity, we can write

$$\begin{aligned} E_n ( t, l ) = f^2 (t) \,\mathcal {E}_n (l) , \quad G_n ( t, l) = f^2(t) \, \hspace{0.5pt}\mathcal {G}_n (l) . \end{aligned}$$

The latter equality makes sense in generalized form also in the case $\hspace{0.5pt}\mathcal {G}_n (l) = +\infty$. In particular, $E_n ( \cdot ,l)$ is of class $C^1 ( [0,T])$ while, by Lemma 1, $E_n ( t, \cdot )$ is decreasing, continuous and of class $C^1( (0,L) {\setminus } \Lambda _n)$. The energy release rate $G_n (\cdot , l)$ is instead of class $C^1 ( [0,T] )$ if $\hspace{0.5pt}\mathcal {G}_n (l ) < + \infty$ (for example if $l \not \in \Lambda _n$), otherwise $G_n (\cdot , l) = + \infty$ for $t>0$ (in this case we will say that $G_n ( \cdot , l)$ is continuous, with extended values). Finally, $G_n ( t , \cdot )$ is right lower semi-continuous in [0, L).

The following proposition provides our formulation of Griffith’s criterion for the quasi-static propagation of a crack in a brittle material with periodic vertical layers:

Proposition 2

There exists a unique non-decreasing, right continuous function $\ell _n : [0,T) \rightarrow [L_0, L]$ which satisfies the initial condition $\ell _n(0) = L_0$ and Griffith’s criterion in the form

• (i) $G_n (t, \ell _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n(t))$ for every time $t \in [0,T)$;

• (ii) if $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t))$ then $\ell _n$ is right differentiable in t and $\dot{\ell }^+_n (t) =0$;

• (iii) if $t \in J ( \ell _n)$ then $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [\ell ^-_n(t) , \ell _n(t) )$,

where $J (\ell _n)$ denotes the set of discontinuity points of $\ell _n$.

Before proving Proposition 2 let us make a few comments. The uniqueness is true, thanks to the monotonicity of the boundary condition together with the right continuity of the evolution. The discontinuity points $t \in J(\ell _n)$ correspond to instantaneous non-equilibrium transitions of the system between the equilibria $\ell _n^- (t)$ and $\ell _n(t) = \ell _n^+(t)$. Often discontinuities are tailored to the layers: roughly speaking, the crack jumps through the layer with smaller toughness, reaching the closest interface, where it “re-nucleates” before advancing in the layer with higher toughness. In this case the non-equilibrium condition $G_n (t, l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ may not hold for $l = \ell _n^+(t) \in \Lambda _n$. For instance, it may happen that $G_n (t, l) > G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}$ for $l \in (l_{n,k} , l_{n,k+\lambda } )$ while $G_n (t, l_{n,k+\lambda } ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$. For this reason condition (ii) holds with the right derivative ${\dot{\ell }}_n^+(t)$; however, out of interfaces the full time derivative ${\dot{\ell }}_n (t)$ can be used (see Corollary 4). However, this is neither a general rule for the evolution neither an a-priori assumption in our analysis; in some cases the crack may as well cross multiple layers in a single discontinuity, depending on the parameters.

Strictly speaking, right continuity is not necessary for Griffith’s criterion, however it is also not a restrictive since $\ell _n$ turns out to be the (only) right continuous representative of any evolution $\lambda _n$ which satisfies Griffith’s criterion. More precisely, we have the following result, whose proof is postponed after Corollary 4:

Corollary 3

If $\lambda _n : [0,T] \rightarrow [L_0,L]$ is non-decreasing, satisfies $\lambda _n (0) = L_0$ and Griffith’s criterion, that is,

• (i) $G_n (t, \lambda _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \lambda _n(t))$ for every time $t \in [0,T]$;

• (ii) if $G_n ( t, \lambda _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \lambda _n (t))$ and $t<T$ then $\lambda _n$ is right differentiable in t and ${\dot{\lambda }}^+_n (t) =0$;

• (iii) if $t \in J ( \lambda _n)$ then $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [\lambda ^-_n(t) , \lambda ^+_n(t) )$,

then the right continuous representative of $\lambda _n$ coincides with $\ell _n$ in [0, T).

In other terms, $\{ t \in [0,T] : \lambda _n (t) \ne \ell _n (t) \} \subset J (\ell _n) = J (\lambda _n)$. Roughly speaking, if $\lambda _n (t) \ne \ell _n(t)$, then t is a jump point, where $\lambda _n (t) = \ell _n^-(t)$ and $\lambda _n^+ (t) = \ell _n (t)$.

Proof of Proposition 2

The proof is based on the following explicit representation of the evolution:

$$\begin{aligned} \ell _n (t) = \inf \{ l \in [L_0,L] : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} . \end{aligned}$$

(7)

Note that the set $\{ l \in [L_0,L] : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \}$ is not empty because, by definition, $G_n (t, L) =0$. Thus $\ell _n$ is well defined and takes values in $[L_0,L]$. We will check that the function $\ell _n$ given by (7) is indeed the unique solution.

Clearly $\ell _n$ satisfies the initial condition because $G_n ( 0 , L_0) = 0$.

Let us check monotonicity. Since the function f is increasing and $G_n$ is non-negative, for every $0 \leqq t_1< t_2 < T$, it holds that $G_n ( t_{\tiny \text {1}}, l ) = f^2(t_{\tiny \text {1}}) \hspace{0.5pt}\mathcal {G}_n ( l) \leqq f^2(t_{\tiny \text {2}}) \hspace{0.5pt}\mathcal {G}_n (l) = G_n ( t_{\tiny \text {2}}, l )$, and hence

$$\begin{aligned} \{ l \in [L_0,L] : G_n ( t_{\tiny \text {1}}, l)< G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} \supset \{ l \in [L_0,L] : G_n ( t_{\tiny \text {2}}, l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} \,. \end{aligned}$$

Taking the infimum yields that $\ell _n (t_{\tiny \text {1}}) \leqq \ell _n (t_{\tiny \text {2}})$.

Next, we show that $\ell _n$ is right continuous. Since in general $G_n ( \cdot , l)$ is continuous (with extended values) only in (0, T), we distinguish between $t=0$ and $t>0$. In the former case, let $L_0 < l$ with $l \not \in \Lambda _n$. Then as $\tau \rightarrow 0$ we have $G_n ( \tau , l) \rightarrow 0 < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$. Hence, (7) implies $\ell _n (\tau ) \leqq l$ for $\tau > 0$ sufficiently small. By monotonicity $\ell _n^+(0) \leqq l$ and thus $\ell _n^+ (0) \leqq L_0$ by the arbitrariness of l. Let $t > 0$. By monotonicity $\ell ^+_n (t) \leqq \lim _{\tau \rightarrow t^+} \ell _n (\tau )$. Let us prove the opposite inequality. If $\lim _{\tau \rightarrow t^+} \ell _n (\tau ) = L_0$ there is nothing to prove. Otherwise, let $L_0 \leqq l^* < \lim _{\tau \rightarrow t^+} \ell _n(\tau )$ with $l^* \not \in \Lambda _n$ (remember that $\Lambda _n$ is a discrete set). By monotonicity, $l^* < \ell _n (\tau )$ for every $\tau > t$, and thus, applying the definition (7) to $\ell _n(\tau )$, we know that $G_n (\tau , l^*) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l^*)$ for every $\tau > t$. Passing to the limit as $\tau \rightarrow t^+$ by the continuity of $G_n ( \cdot , l^*)$ it follows that $G_n (t, l^*) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n(l^*)$ for every $L_0 \leqq l^* < \lim _{\tau \rightarrow t^+} \ell (\tau )$ with $l^* \not \in \Lambda _n$. Hence $\ell _n(t) = \inf \{ l \in [L_0,L] : G_n ( t, l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n(l) \} \geqq l^*$. Taking the supremum in the right hand side for $L_0 \leqq l^* < \lim _{\tau \rightarrow t^+} \ell (\tau )$ with $l^* \not \in \Lambda _n$, we get $\ell _n (t) \geqq \lim _{\tau \rightarrow t^+} \ell (\tau )$.

Let us check (i). If $\ell _n(t) < L$ then by (7) there exists a sequence $l_m \searrow \ell _n(t)$ such that $G_n (t, l_m) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_m)$ and $l_m \not \in \Lambda _n$. By the right lower semi-continuity of $G_n (t, \cdot )$ and the right continuity of $G^{\hspace{-2.5pt}{ \textit{c}}}_n$ it follows that

$$\begin{aligned} G_n ( t , \ell _n (t) ) \leqq \liminf _{m \rightarrow \infty } G_n (t, l_m ) \leqq \lim _{m \rightarrow \infty } G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_m) = G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \ell _n (t)). \end{aligned}$$

(8)

If $\ell _n(t) =L$ there is nothing to prove since, by definition, $0 = G_n (t , L ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (L)$.

Next we check (ii). Again, we consider separately $t=0$ and $t>0$. If $t=0$ we have $G_n (0, L_0) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (L_0)$. If $\ell _n (\tau ) = L_0$ for some $\tau >0$ then $\ell _n$ is constant up to $\tau$ and thus ${\dot{\ell }}_n^+ (0)=0$. On the contrary, assume that $\ell _n (\tau ) > L_0$ for every $\tau >0$. By (7) we know that $G_n (\tau , l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $\tau >0$ and every $L_0 \leqq l < \ell _n(\tau )$. Hence $f^2(\tau ) \hspace{0.5pt}\mathcal {G}_n ( l ) \geqq \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$, and thus

$$\begin{aligned} f^2 (\tau ) \big ( \mathcal {E}_n (L_0) - \mathcal {E}_n ( \ell _n(\tau ) \big ) = f^2(\tau ) \int _{L_0}^{\ell _n(\tau )} \hspace{0.5pt}\mathcal {G}_n ( l) \, \textrm{d}l \geqq ( \ell _n ( \tau ) - L_0 ) \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} . \end{aligned}$$

As a consequence

$$\begin{aligned} \lim _{\tau \rightarrow 0^+} \frac{\ell _n ( \tau ) - \ell _n(0)}{\tau } \leqq \lim _{\tau \rightarrow 0^+} \frac{ f^2(\tau ) - f^2(0) }{\tau } \frac{ \mathcal {E}_n (L_0) - \mathcal {E}_n ( \ell _n(\tau ))}{ \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } = 0 , \end{aligned}$$

where we used the continuity of the energy $\mathcal {E}_n$ and the fact that $f \in C^1([0,T])$ with $f(0)=0$. Now, consider $t>0$. If $G_n (t, \ell _n(t)) < G_n^c( \ell _n(t))$ then by continuity of $G_n ( \cdot , l)$ we get $G_n ( \tau , \ell _n(t)) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n(t))$ for $\tau$ in a (sufficiently small) right neighborhood of t. By (7) it follows that $\ell _n (\tau ) \leqq \ell _n (t)$; by monotonicity $\ell _n$ is constant in a right neighborhood of t.

Let us consider (iii). Let $t \in J (\ell _n)$. By right continuity $\ell _n(t) > L_0$. By definition $\ell _n (t) = \inf \{ l \in [L_0, L] : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n(l) \}$, hence $G_n (t, l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $L_0 \leqq l < \ell _n (t)$ and in particular for $l \in [ \ell _n^- (t) , \ell _n (t) )$.

To conclude, let us prove uniqueness. Assume, by contradiction, that there exists another right continuous evolution $\ell _* \ne \ell _n$ which satisfies Griffith’s criterion. First, we claim that there exists a time $t \not \in J ( \ell _*)$ such that $\ell _* (t) \ne \ell _n (t)$; indeed, if $\{ t \in [0,T] : \ell _* (t) \ne \ell _n (t) \} \subset J ( \ell _*)$ then by right continuity of $\ell _*$ and $\ell _n$ we would get $\ell _* = \ell _n$ everywhere in [0, T). So, let $t>0$ be a continuity point of $\ell _*$ with $\ell _* (t) \ne \ell _n (t)$.

If $\ell _* (t) > \ell _n (t)$, then, by the definition (7) of $\ell _n$, there exists $l_* \in ( \ell _n (t) , \ell _*(t))$ with $l_* \not \in \Lambda _n$ such that $G_n ( t , l_* ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_*)$. Then, by continuity of the energy release, there exists a neighborhood $[l',l'']$ of $l_*$, with $L_0 \leqq l'$ and $l'' < \ell _*(t)$, such that $G_n ( t , l ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [l',l'']$. Let us see that this interval is a “barrier” for the evolution $\ell _*$. First, note that, by monotonicity in time, $G_n ( \tau , l ) \leqq G_n (t , l ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $\tau \leqq t$ and every $l \in [l',l'']$. Moreover, $\ell _*$ takes all the values in the interval $[l',l'']$, indeed if $l \in [ \ell _*^- ( \tau ) , \ell _*( \tau ))$ for some $\tau \leqq t$ and some $l \in [l', l'']$ then, by condition (iii) of Griffith’s criterion, $G_n (\tau , l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$, which stands in contradiction with $G_n ( \tau , l ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$. Therefore we can define $t_{\tiny \text {1}}= \inf \{ \tau : l' \leqq \ell _* ( \tau ) \}$ and $t_{\tiny \text {2}}= \sup \{ \tau : \ell _* (\tau ) \leqq l'' \}$. It follows that $G_n ( \tau , \ell _* (\tau ) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $\tau \in [ t_{\tiny \text {1}}, t_{\tiny \text {2}})$ and thus, by condition (ii) of Griffith’s criterion, $\ell _*$ is constantly equal to $l'$ in $[t_{\tiny \text {1}}, t_{\tiny \text {2}})$. If $t_{\tiny \text {2}}= t$ then (being t a continuity point) we get $\ell _* (t) = l' < \ell _* (t)$ . If $t_{\tiny \text {2}}< t$ then $\ell _*^- (t_{\tiny \text {2}}) = l'$ while $\ell _*^+ (t_{\tiny \text {2}}) \geqq l''$ and thus $t_{\tiny \text {2}}\in J ( \ell _*)$, this is again a contradiction since Griffith’s criterion implies that $G_n (t_{\tiny \text {2}}, l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [l' , l'')$.

Let $\ell _* (t) < \ell _n (t)$. By the definition of $\ell _n$ we know that $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) > 0$ for every $l \in [ \ell _*(t) , \ell _n (t) )$. By monotonicity $G_n (\tau , l) > G^{\hspace{-2.5pt}{ \textit{c}}}_n(l)$ for every $\tau > t$ and every $l \in [ \ell _*(t) , \ell _n (t) )$. If $\tau \rightarrow t^+$ then $\ell _* (\tau ) \rightarrow \ell _*(t)$ and thus $\ell _* (\tau ) \in [ \ell _*(t) , \ell _n (t) )$ for every time $\tau$ sufficiently close to t; then we have $G_n (\tau , \ell _*(\tau ) ) > G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _*(\tau ))$, which is a contradiction with Griffith’s criterion. $\square$

In order to prove the energy identity we will need also the following corollary, which refines condition (ii) for $\ell _n (t) \not \in \Lambda _n$:

Corollary 4

If $\ell _n(t) \not \in \Lambda _n$ and $G_n ( t , \ell _n (t)) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \ell _n (t))$ then $\ell _n$ is constant in a (sufficiently small) neighborhood of t.

Proof

Let $\ell _n(t) \in (l_{n,k} , l_{n,k+\lambda })$. The energy $E_n$ is of class $C^1$ in $[0,T] \times (l_{n,k} , l_{n,k+\lambda })$ and thus the energy release $G_n$ is continuous. Moreover, $G^{\hspace{-2.5pt}{ \textit{c}}}_n$ is constant in $(l_{n,k} , l_{n,k+1})$. Hence, there exists $\delta >0$ (sufficiently small) such that $G_n ( \tau , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ whenever $| \tau - t | < \delta$ and $| l - \ell _n(t) | < \delta$. By condition (iii) in Proposition 2 it follows that $t \not \in J(\ell _n)$. By continuity of $\ell _n$ let us choose $\tau ' < t$ such that $| \tau ' - t | < \delta$ and $| \ell _n(\tau ') - \ell _n(t) | < \delta$. Then, $G_n ( \tau , \ell _n (\tau ') ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (\tau '))$ for every $\tau \in [ \tau ' , t]$. By Proposition 2 it follows that $\ell _n( \tau ) = \ell _n(t)$ is the unique solution for $\tau \in [ t , t + \delta ]$. The same argument applies if $\ell _n(t) \in (l_{n,k+\lambda } , l_{n,k+1})$. $\square$

Proof of Corollary 3

By uniqueness it is enough to check that the right continuous representative of $\lambda _n$, here denoted by ${\hat{\lambda }}_n$, satisfies Griffith’s criterion, in the sense of Proposition 2.

Arguing by contradiction it is easy to check that $\lambda _n$ is right continuous in 0; indeed, if $\lim _{\tau \rightarrow 0^+} \lambda _n (\tau ) > L_0$ then $0 \in J ( \lambda _n)$ and thus $G_n ( 0 , l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) > 0$, however $G_n ( 0 , l ) = 0$. Hence ${\hat{\lambda }}_n (0) = L_0$.

Let us check condition (i). For $t=0$ there is nothing to prove. For $t>0$ we have $G_n ( \tau , \lambda _n (\tau ) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \lambda _n (\tau ) )$ for every $\tau >t$ and thus the regularity of energy release and toughness imply that

$$\begin{aligned} G_n (t , {\hat{\lambda }}_n (t) ) \leqq \liminf _{\tau \rightarrow t^+} G_n (\tau , \lambda _n (\tau )) \leqq \lim _{\tau \rightarrow t^+} G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \lambda _n (\tau )) = G^{\hspace{-2.5pt}{ \textit{c}}}_n ( {\hat{\lambda }}_n (t)) . \end{aligned}$$

Let us consider (iii). Clearly, $J ( \lambda _n) = J ( {\hat{\lambda }}_n)$ and ${\hat{\lambda }}^\pm _n (t) = \lambda ^\pm _n (t)$. Hence the condition $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [ {\hat{\lambda }}^-_n(t) , {\hat{\lambda }}^+_n(t) )$ holds.

Finally, let us check (ii). If $t \not \in J ( \lambda _n) = J ( {\hat{\lambda }}_n)$ then $\lambda _n (t) = {\hat{\lambda }}_n (t)$ and $\dot{\lambda }_n^+ (t) = 0$ coincides with the right derivative of $\hat{\lambda }_n(t)$, hence there is nothing to check. If $t \in J ( \lambda _n) = J ( {\hat{\lambda }}_n)$ we argue in the following way. We have $G_n (t , {\hat{\lambda }}_n (t)) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( {\hat{\lambda }}_n (t))$, then for $\varepsilon > 0$ sufficiently small there exists a sequence $l_k \searrow {\hat{\lambda }}_n (t)$ such that $l_k \not \in \Lambda _n$, $G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_k) = G^{\hspace{-2.5pt}{ \textit{c}}}_n ( {\hat{\lambda }}_n (t))$ is constant and

$$\begin{aligned} G_n (t, l_k) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_k) - \varepsilon \quad \text {for every }k \in \mathbb {N}. \end{aligned}$$

Writing $G_n ( t , l_k) = f^2 (t) \hspace{0.5pt}\mathcal {G}_n (l_k)$, by time continuity for every $0< \varepsilon ' < \varepsilon$ there exists $\delta >0$ such that

$$\begin{aligned} G_n (\tau , l_k) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_k) - \varepsilon ' \quad \text {for every }k \in \mathbb {N}\text { and every }\tau \in ( t , t + \delta ). \end{aligned}$$

Since $l_k \not \in \Lambda _n$ the energy release $\hspace{0.5pt}\mathcal {G}_n$ is continuous, and thus for every $k \in \mathbb {N}$ there exists $l_k^\flat< l_k< l_k^\sharp < l_{k+1}$ such that

$$\begin{aligned} G_n (\tau , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \quad \text {for every }l \in (l_k^\flat , l_k^\sharp )\text { and every }\tau \in ( t , t + \delta ). \end{aligned}$$

As in proof of Proposition 2 the evolution $\lambda _n$ cannot take value in the intervals $(l^\flat _k, l_k^\sharp )$ for $\tau \in ( t , t + \delta )$. Indeed, there are no jumps, moreover, if $\lambda _n (\tau ') \in (l^\flat _k, l_k^\sharp )$ then condition (ii) implies that the right derivative vanishes and thus $\lambda _n$ is constant in $(\tau ', t + \delta )$. Hence by monotonicity $\lambda _n (\tau ) \leqq l_k^\sharp$ for every k and every $\tau \in ( t , t + \delta )$; in conclusion ${\hat{\lambda }}_n (\tau ) \leqq \inf l_k^\sharp = {\hat{\lambda }}_n (t)$ for every $\tau \in ( t , t + \delta )$. $\square$

Remark 2

For $l \in \Lambda _n$, the extension (6) of the toughness seems the most natural, since

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) = \lim _{s \rightarrow l^+} G^{\hspace{-2.5pt}{ \textit{c}}}_n (s) . \end{aligned}$$

The extension (5), that is,

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_n (l) = \liminf _{ s \rightarrow l^+} \hspace{0.5pt}\mathcal {G}_n (s), \end{aligned}$$

turns out to be the only one compatible with Griffith’s criterion. In principle, since the existence of $\partial ^+_l \mathcal {E}_n (l)$ is not known when $l \in \Lambda _n$ any value $\hspace{0.5pt}\mathcal {G}_n (l)$ with

$$\begin{aligned} \liminf _{s \rightarrow l^+} \frac{\mathcal {E}_n (s) - \mathcal {E}_n (l)}{s - l} \leqq - \hspace{0.5pt}\mathcal {G}_n (l) \leqq \limsup _{s \rightarrow l^+} \frac{\mathcal {E}_n (s) - \mathcal {E}_n (l)}{s - l} \end{aligned}$$

(9)

could be a good candidate for the extension. To fix the ideas assume that $L_0 \in \Lambda _n$ and let ${\tilde{\hspace{0.5pt}\mathcal {G}}}_n(L_0)$ be another extension of $\hspace{0.5pt}\mathcal {G}_n$ in $L_0$. First of all, (9) implies that $\tilde{\hspace{0.5pt}\mathcal {G}}_n(L_0) \geqq \hspace{0.5pt}\mathcal {G}_n ( L_0)$, indeed, by the mean value theorem we have

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_n (L_0)= & \liminf _{ s \rightarrow L_0^+} \hspace{0.5pt}\mathcal {G}_n (s) \leqq \liminf _{s \rightarrow L_0^+} - \frac{\mathcal {E}_n (s) - \mathcal {E}_n (L_0)}{s - L_0}\\ = & - \limsup _{s \rightarrow L_0^+} \frac{\mathcal {E}_n (s) - \mathcal {E}_n (L_0)}{s - L_0} \leqq \tilde{\hspace{0.5pt}\mathcal {G}}_n(L_0) . \end{aligned}$$

Let us consider

$$\begin{aligned} \tilde{\hspace{0.5pt}\mathcal {G}}_n (L_0) > \hspace{0.5pt}\mathcal {G}_n (L_0) = \liminf _{ s \rightarrow L_0^+} \hspace{0.5pt}\mathcal {G}_n (s) . \end{aligned}$$

Clearly, this condition makes sense only when $\liminf _{ s \rightarrow L_0^+} \hspace{0.5pt}\mathcal {G}_n (s) < +\infty$. Hence $0 < \tilde{\hspace{0.5pt}\mathcal {G}}_n (L_0) \leqq +\infty$. Let

$$\begin{aligned} \tilde{G}_n ( 0 , L_0) = 0 ,\quad \tilde{G}_n ( t , L_0 ) = f^2(t) \, {\tilde{\hspace{0.5pt}\mathcal {G}}}_n(L_0) \ \text {for }t>0. \end{aligned}$$

Hence $\tilde{G}_n (t, L_0) > G_n (t, L_0)$ for $t > 0$. Assume that there exists a quasi-static evolution ${\tilde{\ell }}_n$ which satisfies Griffith’s criterion with $\tilde{G}_n$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_n$. We denote by $t_*$ and $t^*$ the points $t_* = \max \, \{ t \geqq 0 : \tilde{G}_n ( t , L_0) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (L_0) \}$ and $t^* = \max \, \{ t \geqq 0 : G_n ( t , L_0) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n(L_0) \}$. We have $t_* < t^*$. Fix $t' \in ( t_* , t^*)$. By the definition of $\hspace{0.5pt}\mathcal {G}_n (L_0)$ there exists $l_m \searrow L_0$ such that $l_m \not \in \Lambda _n$ and $\hspace{0.5pt}\mathcal {G}_n (l_m) \rightarrow \hspace{0.5pt}\mathcal {G}_n ( L_0) < \tilde{\hspace{0.5pt}\mathcal {G}}_n (L_0)$. Upon extracting a subsequence, not relabeled, we can assume that there exists $t_m < t'$ such that either ${\tilde{\ell }}_n (t_m) = l_m$ for every $m \in \mathbb {N}$ or $l_m \in [ {\tilde{\ell }}^-_n (t_m ) , \ell _n (t_m ) )$ for every $m \in \mathbb {N}$.

In the former case, for every $t \in (t_m , t')$, we have

$$\begin{aligned} G_n ( t , l_m) \leqq G_n ( t' , l_m) \rightarrow G_n ( t', L_0) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( L_0) = G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_m) . \end{aligned}$$

Hence for $m \gg 1$ we have $G_n ( t , l_m) < G^{\hspace{-2.5pt}{ \textit{c}}}( l_m)$ for every $t \in (t_m , t')$. As a consequence ${\tilde{\ell }}_n (t) = l_m$ in $(t_m , t')$. Repeating the argument for $l_{m+1}$ leads to ${\tilde{\ell }}_n (t) = l_{m+1}$ in $(t_{m+1} , t')$ which is a contradiction since $t_{m+1} < t_m$ and $l_{m+1} \ne l_m$.

In the other case, that is, when $l_m \in [ {\tilde{\ell }}^-_n (t_m ) , \ell _n (t_m ) )$ for every $m \in \mathbb {N}$, we argue as follows. As above,

$$\begin{aligned} G_n (t_m , l_m) \leqq G_n (t' , l_m) \rightarrow G_n ( t' ,L_0) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( L_0) = G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_m) \end{aligned}$$

and thus for $m \gg 1$ we have $G_n ( t_m , l_m) < G^{\hspace{-2.5pt}{ \textit{c}}}( l_m)$ which contradicts condition (iii).

In summary, the right lower semi-continuity of the energy release $\hspace{0.5pt}\mathcal {G}_n$ is necessary for the existence of a quasi-static evolution in the sense of Proposition 2.

Finally, note that the representation (7) is independent of the extension of $\hspace{0.5pt}\mathcal {G}_n$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_n$ in $\Lambda _n {\setminus } \{ L \}$; indeed, (7) can also be written as

$$\begin{aligned} \ell _n (t) = \inf \{ l \in [L_0,L] {\setminus } \Lambda _n \text { or } l =L : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} . \end{aligned}$$

2.3 Energy Identity

In the sequel the energy balance will be crucial to explain toughening. To this end, we introduce the potential energy $F : [0,T] \times [L_0,L] \rightarrow \mathbb {R}$ given by

$$\begin{aligned} F_n (t, l) = E_n ( t, l) + D_n (l ) , \end{aligned}$$

where $D_n :[L_0,L] \rightarrow \mathbb {R}$ is the dissipated energy given by

$$\begin{aligned} D_n (l) = \int _{L_0}^l G^{\hspace{-2.5pt}{ \textit{c}}}_n (s) \, \textrm{d}s . \end{aligned}$$

Note that the energy $F_n ( \cdot , l)$ is differentiable in [0, T] while $F_n ( t, \cdot )$ is differentiable only in $[L_0,L] {\setminus } \Lambda _n$.

Corollary 5

Let $\ell _n$ be the quasi-static evolution provided by Proposition 2. For every $t \in [0,T)$ the following energy identity holds:

$$\begin{aligned} F_n ( t , \ell _n(t)) = \int _0^t \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau + \hspace{-12pt} \sum _{\tau \, \in \, J(\ell _n) \cap [0,t]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket . \end{aligned}$$

(10)

Where $\llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket = F_n ( \tau , \ell _n (\tau )) - F_n ( \tau , \ell _n^- (\tau )) \leqq 0$.

We recall that $F_n ( 0 , L_0 )=0$. The energy identity (10) can be written also as

$$\begin{aligned} E_n (t, \ell _n(t)) = \int _0^t P_n ( t, \ell _n (t)) \, \textrm{d}t - D_n ( \ell _n(t)) + \hspace{-12pt} \sum _{\tau \, \in \, J(\ell _n) \cap [0,t]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket , \end{aligned}$$

(11)

where $P_n ( t, \ell _n (t)) = \partial _t F ( t , l) = \partial _t E ( t, l)$ is the power of “external forces”. Therefore, part of the energy (supplied by the work of external forces) is stored in the elastic energy, part is dissipated by the crack, while part is dissipated in the discontinuity points. In this respect, we recall that the rate-independent evolutions $\ell _n$ can be obtained also from rate-dependent evolutions (see for example [14]) by vanishing viscosity; in this approach, the energy dissipated in the jumps turns out to be the limit of the energy dissipated in the rate-dependent processes.

Clearly, by right continuity the energy identity holds also in every time interval $[t_{\tiny \text {1}}, t_{\tiny \text {2}}]$, in the form

$$\begin{aligned} F_n ( t_{\tiny \text {2}}, \ell _n(t_{\tiny \text {2}})) = F_n ( t_{\tiny \text {1}}, \ell _n(t_{\tiny \text {1}})) \! +\! \int _{t_1}^{t_2} \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau \!+\! \hspace{-12pt} \sum _{\tau \, \in \, J(\ell _n) \cap (t_1,t_2]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket . \end{aligned}$$

Proof

In general the energy $F_n ( t, \cdot )$ is not differentiable in $\Lambda _n$, thus the idea is to consider the sub-intervals in which the evolution takes values in $[L_0,L] {\setminus } \Lambda _n$, taking care of the possible discontinuities. In this way we also make reference to the stick-slip and re-nucleation effect at the interfaces, see [13].

Let $l_{n,k} = k l_n = k L /n$. Let $k_0$ and $k_I$ such that

$$\begin{aligned} l_{n,k_0} \leqq L_0< l_{n,k_1} , \quad l_{n,k_I-1} < \sup \ell _n \leqq l_{n,k_I} . \end{aligned}$$

If $I=0$ we have $\ell _n (t) \equiv L_0$, thus the energy identity (10) boils down to

$$\begin{aligned} F_n (t, \ell _n(t)) = \int _0^t \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau , \end{aligned}$$

which is clearly true by the time regularity of the energy. If $I >0$, consider the points $l_{n,k} \in \Lambda _n$ for $k=k_0, \ldots , k_I$ and define the times

$$\begin{aligned} t_{n,k_0}= & \sup \{ t : \ell _n (t) \leqq L_0 \} , \quad t^\flat _{n,k_I} = \sup \{ t : \ell _n (t)< l_{n,k_I} \} , \quad t^\sharp _{n,k_I} = T , \\ t_{n,k_i}^\flat= & \sup \{ t : \ell _n (t)< l_{n,k_i} \} , \quad t_{n,k_i}^\sharp = \inf \{ t : \ell _n (t) \!>\! l_{n,k_i} \} \quad \text {for }0 \!<\! i \!<\! I\text { (if any).} \end{aligned}$$

In this way we have a finite partition of [0, T] given by

$$\begin{aligned} 0 < t_{n,k_0} \leqq t_{n,k_{\tiny \text {1}}}^\flat \leqq t_{n,k_{\tiny \text {1}}}^\sharp \leqq \cdots \leqq t_{n,k_I}^\flat \leqq t_{n,k_I}^\sharp = T ; \end{aligned}$$

note that some of these points may coincide. We will prove (10) by induction, for $i=0,\ldots ,I$.

In the interval $[0,t_{n,k_0})$ we have $\ell _n (t) = L_0$ . Thus for every $t \in [0,t_{n,k_0})$ we can write

$$\begin{aligned} F_n (t ,\ell _n(t) ) = \int _0^t \partial _t F_n (\tau , \ell _n (\tau )) \, \textrm{d} \tau . \end{aligned}$$

Passing to the limit as $t \nearrow t_{n,k_0}$ we get

$$\begin{aligned} F_n (t_{n,k_0} ,\ell _n^- (t_{n,k_0}) ) = \int _0^{t_{n,k_0}} \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d} \tau , \end{aligned}$$

and then, by right continuity,

$$\begin{aligned} F_n (t_{n,k_0} ,\ell _n (t_{n,k_0}) ) = \int _0^{t_{n,k_0}} \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d} \tau + \llbracket F_n ( t_{n,k_0} , \ell _n (t_{n,k_0} )) \rrbracket . \end{aligned}$$

(12)

If $t^\flat _{n,k_1} > t_{n,k_0}$ then $\ell _n (t) \in ( L_0 , l_{n,k_1} ) \subset ( l_{n,k_0} , l_{n,k_1})$ for $t \in ( t_{n,k_0} , t^\flat _{n,k_1})$. In the interval $(L_0, l_{n,k_1})$ the elastic energy $\mathcal {E}_n ( t, \cdot )$ is of class $C^1$ while the dissipated energy $D_n$ is affine, since $G_n^c$ is constant. Thus in every subinterval $t_{n,k_0}< t'< t < t^\flat _{n,k_1}$ we can apply the chain rule in BV to get

$$\begin{aligned} \begin{aligned} F_n (t , \ell _n (t ) )&= F_n (t', \ell _n(t')) \!+\! \int _{t'}^{t} \partial _t F_n (\tau , \ell _n (\tau ) ) \, \text {d} \tau \!+\! \int _{t'}^{t} \partial _l F_n ( \tau , \ell _n (\tau ) ) \, \text {d}_{LC} \ell _n (\tau ) \\ &\quad + \sum _{ \tau \in J (\ell _n) \cap (t', t ]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket , \end{aligned} \end{aligned}$$

where $\text {d}_{LC} \ell _n$ denotes the sum of the absolutely continuous and Cantor part of the measure $\text {d} \ell _n$. Note that $\text {d}_{LC} \ell _n$ is concentrated on the set of continuity points of $\ell _n$. We claim that

$$\begin{aligned} \begin{aligned} \int _{t'}^{t} \partial _l F_n ( \tau , \ell _n (\tau ) ) \, \text {d}_{LC} \ell _n (\tau ) = 0 . \end{aligned} \end{aligned}$$

Indeed, for $l \in ( L_0 , l_{n,k_1} )$ we have $\partial _l F_n ( \tau , l ) = - G_n (\tau , l ) + G_n^c(l)$. By Proposition 2 we known that $G_n (\tau , \ell _n(\tau ) ) \leqq G_n^c(\ell _n(\tau ))$. If equality holds then $\partial _l F_n ( \tau , \ell _n (\tau ) ) = 0$. If $G_n (\tau , \ell _n(\tau ) ) < G_n^c(\ell _n(\tau ))$ then by Corollary 4 it follows that $\ell _n$ is constant in a neighborhood of $\tau$, and thus $\text {d}_{LC} \ell _n (\tau ) = 0$. The claim is proved. Now, passing to the limit as $t'\rightarrow t_{n,k_0}$ and using (12) we get

$$\begin{aligned} \begin{aligned} F_n (t , \ell _n ( t ) )&= F_n ( t_{n,k_0} , \ell ( t_{n,k_0} ) ) + \int _{t_{n,k_0} }^{t} \partial _t F_n (\tau , \ell _n (\tau ) ) \, \text {d} \tau \\ &\quad + \sum _{ \tau \in J (\ell _n) \cap (t_{n,k_0} , t ] } \llbracket F_n ( \tau , \ell (\tau )) \rrbracket \\ &= \int _{0}^{t} \partial _t F_n (\tau , \ell _n (\tau ) ) \, \text {d} \tau + \sum _{ \tau \in J (\ell _n) \cap [0 , t ] } \llbracket F_n ( \tau , \ell (\tau )) \rrbracket . \end{aligned} \end{aligned}$$

(13)

Passing to the limit as $t \rightarrow t_{n,k_1}^\flat$ and taking into account the possible jump in $t_{n,k_1}^\flat$ yields

$$\begin{aligned} F_n (t_{n,k_1}^\flat , \ell _n ( t_{n,k_1}^\flat ) ) = \int _{0}^{t_{n,k_1}^\flat } \partial _t F_n (\tau , \ell _n (\tau ) ) \, \textrm{d} \tau + \sum _{ \tau \in J (\ell _n) \cap [0 , t_{n,k_1}^\flat ] } \llbracket F_n ( \tau , \ell (\tau )) \rrbracket . \end{aligned}$$

(14)

If $t^\flat _{n,k_1} = t_{n,k_0}$ there is nothing to prove since the previous identity boils down to (12).

If $t^\flat _{n,k_1} < t^\sharp _{n,k_1}$ then $\ell _n = l_{n,k_1}$ in the interval $(t^\flat _{n,k_1} , t^\sharp _{n,k_1} )$, thus for every $t \in ( t^\flat _{n,k_1} , t^\sharp _{n,k_1} )$ we can write

$$\begin{aligned} F (t , \ell _n ( t) ) =&\int _{0}^{t} \partial _t F_n (\tau , \ell _n (\tau ) ) \, \textrm{d} \tau + \sum _{ \tau \in J (\ell _n) \cap [0 , t ] } \llbracket F_n ( \tau , \ell (\tau )) \rrbracket . \end{aligned}$$

(15)

If $t^\flat _{n,k_1} = t^\sharp _{n,k_1}$ then (14) holds replacing $t^\flat _{n,k_1}$ with $t^\sharp _{n,k_1}$. We proceed in this way up to any time $t <T$. $\square$

2.4 Compactness and Convergence of Evolutions

By Helly’s Theorem it is well known that up to non-relabeled subsequences the evolutions $\ell _n$ converge to a certain limit $\ell$. The goal of the next proposition is to provide a first characterization of the limit $\ell$ independently of the convergence of the energies.

Proposition 6

Let $\ell _n$ be the sequence of quasi-static evolutions given by Proposition 2. There exists a subsequence (not relabeled) such that $\ell _n \rightarrow \ell$ pointwise in [0, T). For $t\in [0,T)$ the right continuous representative of the limit $\ell$ is characterized by

$$\begin{aligned} \ell ^+ (t) = \inf \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} , \end{aligned}$$

(16)

where $1 / f^2 (0) = +\infty$.

Proof

We recall that

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) = \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l_n)}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} : l_n \rightarrow l \Big \} . \end{aligned}$$

For the theory of $\Gamma$-convergence we refer the reader to [10].

Let $t > 0$. By Proposition 2 we know that $G_n (t , \ell _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \ell _n(t) )$, hence $G_n (t , \ell _n(t) )$ is finite. Writing $G_n (t , \ell _n(t) ) = f^2 (t) \, \mathcal {G}_n ( \ell _n (t ))$ we get

$$\begin{aligned} \frac{1}{f^{\tiny \text {2}}(t)} \geqq \frac{\hspace{0.5pt}\mathcal {G}_n (\ell _n(t))}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n( t) )} . \end{aligned}$$

The above inequality holds also in generalized sense for $t=0$. Since $\ell _n (t) \rightarrow \ell (t)$ pointwise for every $t \in [0,T)$ we have

$$\begin{aligned} \frac{1}{f^2 (t)}\geqq & \liminf _{n \rightarrow \infty } \frac{ \hspace{0.5pt}\mathcal {G}_n (\ell _n(t))}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n( t) )} \geqq \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l_n)}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} : l_n \rightarrow \ell (t) \Big \}\\ = & \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (\ell (t)) . \end{aligned}$$

Let $\tau > t \in [0,T)$. By the monotonicity of f we can write

$$\begin{aligned} \frac{1}{f^2(t)} > \frac{1}{f^2 (\tau )} \geqq \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (\ell (\tau )) . \end{aligned}$$

Hence

$$\begin{aligned} \ell (\tau ) \in \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} \quad \text { for every }\tau > t \end{aligned}$$

and then

$$\begin{aligned} \ell ^+ (t) \geqq \inf \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} . \end{aligned}$$

Let us prove the opposite inequality. If $\ell ^+ (t) = L_0$ there is nothing to prove. Otherwise, let $L_0 \leqq l < \ell ^+(t) \leqq \ell (\tau )$ for $\tau > t$. Let $\tau _n \searrow t^+$ such that $\ell _n (\tau _n) \rightarrow \ell ^+(t)$ and let $l_n$ be a “recovery sequence” for l, that is, $l_n \rightarrow l$ and

$$\begin{aligned} \liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l_n)}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n) } = \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) . \end{aligned}$$

From the representation (7) of the evolution $\ell _n$ we know that

$$\begin{aligned} \ell _n (\tau _n)&= \inf \{ l \in [L_0,L] : G_n ( \tau _n , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} \\&= \inf \left\{ l \in [L_0,L] : \frac{1}{f^2 (\tau _n)} > \frac{\hspace{0.5pt}\mathcal {G}_n ( l) }{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)} \right\} . \end{aligned}$$

Being $l_n < \ell _n (\tau _n)$ for $n \gg 1$ (since $l < \ell ^+(t)$) the above representation implies that

$$\begin{aligned} \frac{1}{f^2 (\tau _n)} \leqq \frac{\hspace{0.5pt}\mathcal {G}_n ( l_n) }{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} . \end{aligned}$$

Taking the liminf as $n \rightarrow \infty$ and remembering that $l_n$ is a recovery sequence, we get

$$\begin{aligned} \frac{1}{f^2 (t)} \leqq \liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l_n)}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} = \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) . \end{aligned}$$

Thus, every $l < \ell ^+ (t)$ does not belong to the set

$$\begin{aligned} \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} \end{aligned}$$

and then

$$\begin{aligned} \ell ^+(t) \leqq \inf \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} , \end{aligned}$$

which concludes the proof. $\square$

Remark 3

The characterization (16) seems to be the natural choice combined with the pointwise convergence of a subsequence (extracted in Proposition 6) of the quasi-static evolutions $\ell _n$. However, if there exists

$$\begin{aligned} \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l), \end{aligned}$$

(17)

then the whole sequence $\ell _n {\mathop {\rightharpoonup }\limits ^{*}}\ell$ in BV(0, T). Indeed, each function $\ell _{n}$ is monotone and bounded in $[L_0,L]$, hence the sequence $\ell _n$ is weakly* pre-compact in BV(0, T). Given any subsequence $\ell _{n_k}$ of $\ell _n$ there exists a further subsequence, denoted by $\ell _{n_i}$, converging to some limit $\ell$ weakly in BV(0, T) and pointwise in (0, T). By Proposition 6 the limit $\ell$ (possibly depending on the subsequence) is characterized by

$$\begin{aligned} \ell ^+ (t)= & \inf \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_{n_i}}{ G^{\hspace{-2.5pt}{ \textit{c}}}_{n_i}} (l)< \frac{1}{f^2 (t)} \right\} \\= & \inf \left\{ l \in [L_0,L] : \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_{n}}{ G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} (l) < \frac{1}{f^2 (t)} \right\} . \end{aligned}$$

Therefore, the evolution $\ell$ is actually independent of the subsequence $\ell _{n_k}$. Clearly, this argument requires the existence of the $\Gamma$-limit in (17) while the $\Gamma$-liminf in (16) is always defined.

2.5 Homogenization of the Elastic Energy

In this section we will first recall a few results on the homogenization of the elastic energy. Let us introduce the homogenized stiffness matrix $\, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ given by

$$\begin{aligned} \, \varvec{C}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} = \left( \begin{matrix} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} \end{matrix} \right) , \end{aligned}$$

where $1 / \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \lambda / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + (1-\lambda ) / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ is the weak* limit of $1/\mu _{n,{\tiny \text {1}}}$ and $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}$ is the weak* limit of $\mu _{n,{\tiny \text {2}}}$. Note that the coefficients $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},i}$ are constant and independent of $l \in (0,L)$. We denote by $\mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ the homogenized elastic energy and by $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ its static condensation. By classical results on the homogenization of elliptic problems (see Appendix A) and on the energy release (see for example [20] and the references therein) we infer the following proposition.

Proposition 7

If $l_n \rightarrow l$ then $\mathcal {E}_n ( l_n) \rightarrow \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. The energy $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is of class $C^1 (0,L)$.

Thanks to the regularity of $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ we can define the energy release

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = - \partial _l^+ \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = - \partial _l \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \quad \text {for }l \in (0,L), \quad \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(L) = 0 , \end{aligned}$$

which is continuous in (0, L). We remark that in general $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is neither the almost everywhere limit nor the $\Gamma$-liminf of $\hspace{0.5pt}\mathcal {G}_n$ (an example is given in §2.8). However, thanks to the regularity and the convergence of the energies we have the following results, which will be useful in the sequel.

Lemma 8

Let $\hspace{0.5pt}\mathcal {G}_{n_i}$ be a subsequence of $\hspace{0.5pt}\mathcal {G}_n$. For almost everywhere $l \in (0,L)$ it holds

$$\begin{aligned} \liminf _{i \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_{n_i} (l) \leqq \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \end{aligned}$$

(Note that the set where the above estimate holds may depend on the subsequence).

Proof

Let $0< l_1< l_2 < L$. By Proposition 7 and by the regularity of the energies we can write

$$\begin{aligned} \int _{l_1}^{l_2} \hspace{0.5pt}\mathcal {G}_{n_i} (s) \, \textrm{d}s = \mathcal {E}_{n_i} (l_1) - \mathcal {E}_{n_i} (l_2) \quad \rightarrow \quad \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l_1) - \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l_2) = \int _{l_1}^{l_2} \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s) \, \textrm{d}s . \end{aligned}$$

Hence, by Fatou’s Lemma we also have

$$\begin{aligned} \int _{l_1}^{l_2} \liminf _{i \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_{n_i} (s) \, \textrm{d}s \leqq \liminf _{i \rightarrow \infty } \int _{l_1}^{l_2} \hspace{0.5pt}\mathcal {G}_{n_i} (s) \, \textrm{d}s = \int _{l_1}^{l_2} \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s) \, \textrm{d}s , \end{aligned}$$

which proves the lower inequality by the arbitrariness of $l_1$ and $l_2$. $\square$

Lemma 9

For every $l \in (0,L)$ there exist a sequence $l^\sharp _n \rightarrow l^+$ such that $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \geqq \limsup _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l^\sharp _n)$.

Proof

By the continuity of $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, for every $\delta >0$ there exists $\varepsilon >0$ such that $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s) < \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) + \delta$ for every $s \in [l, l+\varepsilon )$.

We claim that there exists $m \in \mathbb {N}$ such that for every $n > m$ we can find $l_n \in ( l , l + \varepsilon )$ with $\hspace{0.5pt}\mathcal {G}_n (l_n) < \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) + \delta$; from the claim the existence of $l^\sharp _n$ follows. By contradiction, assume that for every $m \in \mathbb {N}$ there exists $n_m > m$ such that $\hspace{0.5pt}\mathcal {G}_{n_m} (s) \geqq \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) + \delta$ for every $s \in ( l , l + \varepsilon )$, then we can find a subsequence such that $\liminf _{m \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_{n_m} (s) \geqq \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) + \delta > \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s)$ in $( l , l + \varepsilon )$, which is a contradiction with Lemma 8. $\square$

From the previous lemma we get the following corollary.

Corollary 10

If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\leqq \Gamma \text {-}\liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n$ then $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \Gamma \text {-}\lim _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n$.

Proof

By Lemma 9 for every $l \in (0,L)$ we have

$$\begin{aligned} \Gamma \text {-}\limsup _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l) \leqq \limsup _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l_n^\sharp ) \leqq \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \leqq \Gamma \text {-}\liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l) , \end{aligned}$$

which concludes the proof. $\square$

As in the previous sections we consider the energy $E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) = f^2(t) \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. Most important, we define $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) = \partial _l E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) = f^2(t) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$ for every $t \in [0,T]$ and $l\in (0,L)$; note that $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$ is finite for $l \in (0,L)$. Finally, we define $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, L)=0$ for every $t \in [0,T]$.

2.6 Effective Toughness and Homogenization of Griffith’s Criterion

The aim of this section is to provide an effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ in such a way that the limit evolution $\ell$ (given by Proposition 6) satisfies Griffith’s criterion for $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$. Formally, regardless of the regularity of the energies involved, we may define $G^c_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ as the function which makes the following identity true:

$$\begin{aligned} \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}{G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}} (l) = \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^c_n} (l) . \end{aligned}$$

(18)

More precisely, we define $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}: [0,L] \rightarrow [ 0, +\infty ]$ as

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = {\left\{ \begin{array}{ll} {\displaystyle \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) } } & \text { if } \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0 , \\ + \infty & \text { if }\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = 0 . \end{array}\right. } \end{aligned}$$

(19)

In the above definition we assume $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)=+\infty$ when $\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) = 0$; by the proof of Lemma 11 (see below) it follows that $\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < + \infty$. As far as Griffith’s criterion is concerned, the definition of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ when $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = 0$ is in some sense arbitrary, indeed, if the evolution takes, at time t, a value l where $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)=0$ then the inequality $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau , l) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ is satisfied for every $\tau \geqq t$ and every non-negative value of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$, further, the evolution will be constantly equal to l for $\tau \geqq t$ (see Remark 5 for a different definition of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ when $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ vanishes).

Note that $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is independent of time and of the evolution, however in principle it may depend on every parameter and every datum of the problem; in mechanical terms, $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is an R-curve (R standing for resistance, see for example [2]). In §2.8 we will give an example in which $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is actually a constant, depending on the volume fraction $\lambda$, the ratio between the shear moduli $\mu _{{\hspace{-4pt} A},i}$ and $\mu _{{\hspace{-3.5pt} B},i}$ and the toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$. Before proceeding, we provide this lemma.

Lemma 11

The effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is upper semi-continuous (and thus Borel measurable). Moreover, $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}\geqq \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$.

Proof

It is enough to consider the set $\{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\ne 0 \}$ since, by continuity, the set $\{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= 0 \}$ is closed. The upper semi-continuity follows from the continuity of $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and the lower semi-continuity of the $\Gamma \text {-}\liminf$. Next, write

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l)&= \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{ \hspace{0.5pt}\mathcal {G}_n (l_n)}{G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} : l_n \rightarrow l \Big \} \\&\leqq \frac{1}{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} \inf \left\{ \liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l_n) : l_n \rightarrow l \right\} \\&\leqq \frac{\Gamma \text {-}\liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l)}{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} . \end{aligned}$$

By Lemma 8 we infer

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l) \leqq \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \end{aligned}$$

(20)

and thus

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) \leqq \frac{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)}{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} . \end{aligned}$$

(21)

Hence, $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \geqq \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$. $\square$

A local upper bound for $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is given in Corollary 14, as a byproduct of the following theorem:

Theorem 12

Let $\ell$ be the limit evolution given by Proposition 6. Then $\ell ^+ (t) = \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)$, where $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}: [0,T) \rightarrow [L_0, L]$ is the unique non-decreasing, right continuous function which satisfies the initial condition $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(0) = L_0$ and Griffith’s criterion:

• (i) $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t))$ for every time $t \in [0,T)$;

• (ii) if $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t))$ then $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is right differentiable in t and $\dot{\ell }_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^+ (t)=0$;

• (iii) if $t \in J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , s ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(s)$ for every $s \in [\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) )$.

Proof

First, we check that for every $t \in [0,T)$ we have

$$\begin{aligned} \{ l \in [L_0,L] : G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l)< G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \}&= \left\{ l \in [L_0,L] : \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < \frac{1}{f^2 (t)} \right\} . \end{aligned}$$

(22)

If $t=0$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( 0 , l ) = 0$ and thus by Lemma 11 we have $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( 0 , l ) < \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ for every $l \in [L_0,L]$. By (21) we can write

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) < +\infty = \frac{1}{f^2 (0)} \end{aligned}$$

for every $l \in [L_0,L]$. If $t>0$ we write $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, l) = f^2 (t) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$, using the definition of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ we get

$$\begin{aligned} G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l)< G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \quad \Leftrightarrow \quad f^2 (t) < \frac{1}{\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{ \hspace{0.5pt}\mathcal {G}_n }{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) } . \end{aligned}$$

If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = 0$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) = 0$ and thus $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) < \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$. Moreover by (21) we have

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{ G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) = 0 < \frac{1}{f^2 (t)} . \end{aligned}$$

By (22) we can now invoke Proposition 6 which gives the representation

$$\begin{aligned} \ell ^+ ( t) = \inf \{ l \in [L_0,L] : G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \} . \end{aligned}$$

(23)

At this point it is enough to follow the proof of Proposition 2, with a few minor differences, in order to show that the function $\ell ^+$ defined by (23) satisfies Griffith’s criterion. The proof is concluded. $\square$

Remark 4

In general, the limit evolution $\ell$ does not satisfy Griffith’s criterion in terms of the energy release $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, of the homogenized energy $E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, and the homogenized toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ (1-\lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$. Starting from the (16), which is independent of the limit energies, there are actually different ways to proceed. In our approach we define an effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ in such a way that the limit $\ell$ satisfies Griffith’s criterion for $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$. This idea seems very natural in terms of convergence of the energies and is consistent with the approach followed in mechanics, see for example [13]. On the other hand it could be interesting also to fix $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and define an effective energy release $\hspace{0.5pt}\mathcal {G}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ taking the limit of $\hspace{0.5pt}\mathcal {G}_n$. However, even if $\hspace{0.5pt}\mathcal {G}_n$ has a “nice” integral representation (see Lemma 1) it seems difficult to study its limit, the main issue being the fact that the support of the auxiliary functions $\phi$ shrinks with n. In §3.4 we will consider a different setting in which the representation of $\hspace{0.5pt}\mathcal {G}_n$ holds for a function $\phi$ independent of n (see Lemma 21); in that case the limit of $\hspace{0.5pt}\mathcal {G}_n$ can be computed, and actually equals $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$.

2.6.1 Plane-Strain Setting

The definition (19) of effective toughness and the convergence result of Theorem 12 can be easily extended to the plane strain setting with minor changes in the proofs. On the contrary, the generalization of the example and of the results provided in §2.8 and §3 is not straightforward, indeed, the computations provided in that sections are tailored to the anti-plane setting, being based on the specific form of the homogenized energy and of the energy release.

We give a brief outline. Consider the same geometry for the reference domain, the cracks and the layers. Consider a boundary condition of the form f(t)g where $g \in H^1(\partial _D \Omega ; \mathbb {R}^2)$. The condensed elastic energy $\mathcal {E}_n$ is again of class $C^1( (0,L) {\setminus } \Lambda _n)$, thus, as in Proposition 2, there exists a unique, right-continuous evolution $\ell _n$ which satisfies Griffith’s criterion and which can be characterized by $\ell _n (t) = \inf \{ l \in [L_0,L] : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \}$. The regularity of the energy $\mathcal {E}_n$ follows from an integral representation of the energy release, as that of Lemma 1, see for example [20].

Next, by classical results on H-convergence, see for example [11], the linear elastic energy $E_n$ converge to a linear elastic energy $E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ with constant coefficients, in particular the condensed (or reduced) energy $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is of class $C^1(0,L)$.

By Helly’s Theorem, upon extracting a subsequence, there exists a limit evolution $\ell$. After defining the effective toughness, as in (19), and arguing as in Theorem 12 it follows that $\ell ^+ = \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, where $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is again the unique right-continuous evolution which satisfies Griffith’s criterion.

2.6.2 Evolutionary $\varvec{\Gamma }$-Convergence

In this section we give a slight generalization of Theorem 12, in order to provide an evolutionary $\Gamma$-convergence result for our rate independent system, in the spirit of [15]. To this end, let us introduce the rate independent dissipation functionals

$$\begin{aligned} \mathcal {R}_n ( l , \dot{l} ) = {\left\{ \begin{array}{ll} G^{\hspace{-2.5pt}{ \textit{c}}}_n ( l ) \, \dot{l} & \dot{l} \geqq 0 , \\ + \infty & \text {otherwise,} \end{array}\right. } \quad \mathcal {R}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( l , \dot{l} ) = {\left\{ \begin{array}{ll} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \dot{l} & \dot{l} \geqq 0 , \\ + \infty & \text {otherwise.} \end{array}\right. } \end{aligned}$$

We refer the reader to Appendix 4 for the notion of balanced viscosity solution for the rate independent system $(E_n , \mathcal {R}_n)$, in the language of [16]; actually (see again Appendix 4), these solutions coincide with the quasi-static evolutions of Corollary 3. Therefore the right continuous representative of any balanced viscosity solution of the rate independent system $(E_n , \mathcal {R}_n)$ coincides with $\ell _n$, that is, the unique right continuous evolution provided by Proposition 2.

Corollary 13

The rate independent systems $( E_n , \mathcal {R}_n)$ evolutionary $\Gamma$-converge to $( E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, \mathcal {R}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}})$.

Proof

By [15, Definition 2.10] the above statement is equivalent to the following: let $L_n \in (0,L)$ such that $L_n \rightarrow L_0 \in (0,L)$ and let $\gamma _n$ be a solution of the rate independent system $(E_n , \mathcal {R}_n)$ with intial condition $L_n$, then $\gamma _n$ converge to $\gamma _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ where $\gamma _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is a solution of the rate independent system $(E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, \mathcal {R}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}})$ with initial condition $L_0$.

By Helly’s Theorem there exists a subsequence (not relabeled) such that $\gamma _n \rightarrow \gamma$ pointwise in [0, T). In order to characterize the limit $\gamma$ it is not restrictive to consider the convergence of the sequence $\ell _n$ of the right continuous representatives of $\gamma _n$. At this point, in order to apply Theorem 12 it is enough to “fix the initial condition”. There are at least a couple of alternatives. The first is to follow, step by step, the proof of Theorem 12 employing the explicit representation

$$\begin{aligned} \ell _n (t) = \inf \{ l \in [L_n,L] : G_n (t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \} . \end{aligned}$$

The second chance is instead to employ the change of variable $\psi _n: [0,L] \rightarrow [0,L]$ given by

$$\begin{aligned} \psi _n (l) = {\left\{ \begin{array}{ll} l L_0 / L_n & l < L_n , \\ (l - L_n) (L - L_0) / (L-L_n) + L_0 & l \geqq L_n , \end{array}\right. } \end{aligned}$$

and then consider the evolutions ${\tilde{\ell }}_n (t) = \psi _n \circ \ell _n (t)$, which satisfy $\tilde{\ell }_n (0)=L_0$. The (tedious) check of convergence is omitted. $\square$

2.6.3 An Upper Bound

Thanks to Griffith’s criterion, we can provide also a simple on a posteriori estimate on the effective toughness.

Corollary 14

Assume that  $\sup \{ \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) : t \in [0,T) \} > L_0$. Then, the effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is locally bounded, more precisely,

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l ) \leqq f^2(T) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \frac{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} }{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } \end{aligned}$$

for every $l \in \textrm{co} \, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}([0,T))$, the convex envelope of the range $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}([0,T))$.

Proof

The convex envelope coincides with the set $\{ l \in [L_0, L] : l \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)$ for some $t \in [0,T) \}$ and is used to “fill the gaps” in the discontinuity points.

If $l \in [ \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) )$ for some $t \in J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$ then $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \leqq G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l ) \leqq f^2(T) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$.

Otherwise, let $L_0< l' < \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)$ for some $t \in (0,T)$. For any sequence $l'_n \rightarrow l'$, we argue as follows. Clearly, $l'_n < \ell _n (t)$ for $n \gg 1$. Remember that $\ell _n (t) = \inf \{ l \in [L_0, L] : G_n (t, l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \}$, hence $G_n (t, l'_n ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l'_n) \geqq \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$ and thus $\hspace{0.5pt}\mathcal {G}_n ( l'_n) \geqq \min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} / f^2(t)$. As a consequence,

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l')&= \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{ \hspace{0.5pt}\mathcal {G}_n (l'_n)}{G^{\hspace{-2.5pt}{ \textit{c}}}_n (l'_n)} : l'_n \rightarrow l' \Big \} \\&\geqq \frac{1}{\max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} \inf \left\{ \liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l'_n) : l'_n \rightarrow l' \right\} \\&\geqq \frac{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}}{ f^2(t) \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} . \end{aligned}$$

It follows that

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l') \leqq \frac{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} f^2(t) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l') }{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } \leqq f^2 (T) \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l') \frac{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} }{\min \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } . \end{aligned}$$

We proved the inequality for any $l' \in (L_0, \sup \{ \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) : t \in [0,T) \} )$. We conclude by the upper semi-continuity of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$. $\square$

In §2.8 we will see an example in which the effective toughness is strictly greater than $\max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$.

Remark 5

Joining the lower bound of Lemma 11 and the upper bound of Corollary 14 it is clear that the set $\textrm{co} \, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}([0,T))$ and the set $\{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= 0\}$ are disjoint. Therefore, to the purpose of proving Theorem 12 it is enough to define $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ in the set $\{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\ne 0\}$. In other terms, we could provide alternative definitions of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$, which still makes Griffith’s criterion true. For instance, let

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_\infty (l) = \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) } \end{aligned}$$

be defined for $l \in [L_0, L]$ such that $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$. Then we could employ as effective toughness the function $\tilde{G}^{\hspace{-2.5pt}\text { c}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = \inf \{ G^{\hspace{-2.5pt}{ \textit{c}}}(l) \}$ where the infimum is taken over all the upper semi-continuous functions $G^{\hspace{-2.5pt}{ \textit{c}}}: [L_0,L] \rightarrow \mathbb {R}$ with $G^{\hspace{-2.5pt}{ \textit{c}}}\geqq G^{\hspace{-2.5pt}{ \textit{c}}}_\infty$ in the set $\{ \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\ne 0 \}$.

2.6.4 Effective Toughness Under Convergence of the Energy Release

In this subsection we consider a couple of cases in which the energy release $\hspace{0.5pt}\mathcal {G}_n$ converge (in a suitable sense) to the energy release $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and, as a consequence, the effective toughness $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is computed explicitly and equals the maximum between $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$.

Corollary 15

Assume that $\hspace{0.5pt}\mathcal {G}_n \rightarrow \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ locally uniformly in (0, L). If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$ then

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} . \end{aligned}$$

Proof

We recall that

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{\displaystyle \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) } . \end{aligned}$$

By uniform convergence $\hspace{0.5pt}\mathcal {G}_n (l_n) \rightarrow \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$ for every $l_n \rightarrow l$, hence

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l)&= \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{ \hspace{0.5pt}\mathcal {G}_n (l_n)}{G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} : l_n \rightarrow l \Big \} \\ &= \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \inf \Big \{ \liminf _{n \rightarrow \infty } \frac{1}{G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n)} : l_n \rightarrow l \Big \} \\&= \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } , \end{aligned}$$

which concludes the proof. $\square$

The previous result can be slightly generalized as follows:

Corollary 16

Let $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$. Assume that $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \leqq \Gamma \text {-}\liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l)$ and that there exists a “joint recovery sequence” for $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$, that is, $l_n \rightarrow l$ such that

$$\begin{aligned} \limsup _{n \rightarrow \infty } G^{\hspace{-2.5pt}{ \textit{c}}}_n (l_n) = \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} , \quad \limsup _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n ( l_n ) = \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) . \end{aligned}$$

Then $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$.

Proof

Clearly $G^{\hspace{-2.5pt}{ \textit{c}}}_n \leqq \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$ and $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = \Gamma \text {-}\lim _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l)$ by Corollary 10. Hence

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l)&\geqq \frac{1}{\max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}} \inf \big \{ \liminf _{n \rightarrow \infty } \hspace{0.5pt}\mathcal {G}_n (l_n) : l_n \rightarrow l \big \} = \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } . \end{aligned}$$

By $\Gamma$-convergence we know that $\hspace{0.5pt}\mathcal {G}_n (l_n) \rightarrow \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. Hence

$$\begin{aligned} \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) \leqq \liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n (l_n)}{G^{\hspace{-2.5pt}{ \textit{c}}}_n(l_n)} = \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } . \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{ \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} } = \Gamma \text {-}\liminf _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_n} (l) , \end{aligned}$$

which concludes the proof. $\square$

2.7 Energy Identity

Given (19) the effective dissipated energy is defined by

$$\begin{aligned} D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( l) = \int _{L_0}^l G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(s) \, \textrm{d}s \end{aligned}$$

and the potential energy by $F (t, l) = E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) + D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$. By Corollary 14 it turns out that $D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}\in AC ( L_0, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(T) )$. Note that $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is Borel measurable since it is upper semi-continuous.

Corollary 17

For every $t \in [0,T)$ the following energy identity holds:

$$\begin{aligned} F ( t , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)) = \int _0^t \partial _t F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \, \textrm{d}\tau + \hspace{-12pt} \sum _{\tau \, \in \, J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap [0,t]} \llbracket F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \rrbracket . \end{aligned}$$

(24)

There $J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$ and $\llbracket F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \rrbracket = F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) - F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^- (\tau )) \leqq 0$.

Proof

In this case we cannot employ the proof of Corollary 5. Clearly, if $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) \equiv L_0$ the proof is trivial by time regularity. As $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\in BV(0,T)$ we can write

$$\begin{aligned} \begin{aligned} \text {d} \ell _{ \textit{hom}\hspace{0.2pt}}= \dot{\ell }_{ \textit{hom}\hspace{0.2pt}}\text {d}\tau + \text {d}_{C} \ell _{ \textit{hom}\hspace{0.2pt}}+ \text {d}_{J} \ell _{ \textit{hom}\hspace{0.2pt}}, \end{aligned} \end{aligned}$$

where $\dot{\ell }_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\in L^1 (0,T)$ denotes the density of the absolutely continuous part (with respect to the Lebesgue measure $\textrm{d}\tau$), $\text {d}_{C} \ell$ denotes the Cantor part, while

$$\begin{aligned} \begin{aligned} \text {d}_{J} \ell _{ \textit{hom}\hspace{0.2pt}}= \sum _{\tau \in J_{ \textit{hom}\hspace{0.2pt}}} \llbracket \ell _{ \textit{hom}\hspace{0.2pt}}(\tau ) \rrbracket \, \delta _{\tau } . \end{aligned} \end{aligned}$$

For convenience, denote $\text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}= {\dot{\ell }}_{ \textit{hom}\hspace{0.2pt}}\text {d}\tau + \text {d}_{C} \ell$.

Being $E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ of class $C^1$ by the chain rule in BV, see for example [1], it follows that

$$\begin{aligned} \begin{aligned} E_{ \textit{hom}\hspace{0.2pt}}(t , \ell _{ \textit{hom}\hspace{0.2pt}}(t))&= E_{\textit{hom}\hspace{0.2pt}}( 0 , \ell _{ \textit{hom}\hspace{0.2pt}}(0) ) + \int _{0}^{t} \partial _t E_{ \textit{hom}\hspace{0.2pt}}( \tau , \ell _{\textit{hom}\hspace{0.2pt}}(\tau ) ) \, \text {d}\tau \\ &\quad + \int _{0}^{t} \partial _l E_{\textit{hom}\hspace{0.2pt}}( \tau , \ell _{ \textit{hom}\hspace{0.2pt}}(\tau ) ) \, \text {d}_{LC} \ell _{\textit{hom}\hspace{0.2pt}}+ \hspace{-10pt} \sum _{\tau \, \in \, J_{ \textit{hom}\hspace{0.2pt}}\cap [0, t] } \hspace{-10pt} \llbracket E_{ \textit{hom}\hspace{0.2pt}}( \tau , \ell _{\textit{hom}\hspace{0.2pt}}(\tau ) ) \rrbracket . \end{aligned} \end{aligned}$$

(25)

By Griffith’s criterion, it remains to prove that a similar representation holds also for the effective dissipated energy, i.e,

$$\begin{aligned} \begin{aligned} D_{\hspace{-2pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}( \ell _{\textit{hom}\hspace{0.2pt}}(t) )&= D_{\hspace{0.15pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}( \ell _{ \textit{hom}\hspace{0.2pt}}(0) ) + \int _{0}^{t} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}( \ell _{ \textit{hom}\hspace{0.2pt}}(\tau ) ) \, \text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}+ \hspace{-12pt} \sum _{\tau \, \in \, J_{ \textit{hom}\hspace{0.2pt}}\cap [0,t] } \hspace{-6pt} \llbracket D ( \tau , \ell _{ \textit{hom}\hspace{0.2pt}}(\tau ) ) \rrbracket . \end{aligned} \end{aligned}$$

(26)

Remember that $D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is not of class $C^1$, however, by definition we can write

$$\begin{aligned} D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l_{\tiny \text {2}}) = D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l_{\tiny \text {1}}) + \int _{l_1}^{l_2} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l , \end{aligned}$$

(27)

for every $0< l_{\tiny \text {1}}< l_{\tiny \text {2}}< L$. Hence

$$\begin{aligned} D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) = \int _{L_0}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l = \int _{L_0}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l_{|R} + \int _{L_0}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l_{|U} , \end{aligned}$$

where $R = \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( [0,T) )$ while $U = \textrm{co} \, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}([0,T)) {\setminus } R$. We will check that

$$\begin{aligned} \int _{L_0}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l_{| U} = \sum _{\tau \, \in \, J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap [0, t ] } \hspace{-6pt} \llbracket D (\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) ) \rrbracket \end{aligned}$$

(28)

and that

$$\begin{aligned} \begin{aligned} \int _{L_0}^{\ell _{\textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}(l) \, \text {d}l_{|R} = \int _{0}^{t} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0pt} \textit{f}\hspace{1pt}}( \ell _{ \textit{hom}\hspace{0.2pt}}(\tau )) \, \text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}} , \end{aligned} \end{aligned}$$

(29)

which, put together, imply (26).

By the right continuity of $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$

$$\begin{aligned} U = \bigcup _{\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} U_\tau , \end{aligned}$$

(30)

where $U_\tau$ denotes an interval of the form $(\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) )$ or $[\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) )$ (the fact that $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )$ is excluded is due to the right continuity of $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$). Note that the sets $U_\tau$ are pairwise disjoint and remember that $J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is a countable set. Then

$$\begin{aligned} \int _{L_0}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l_{| U}&= \sum _{\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap [0,t]} \int _{\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l \\ &= \sum _{\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap [0,t]} D ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) ) - D (\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) ) . \end{aligned}$$

In order to prove (29), we will first check that $\text {d}l_{|R}= \text {d}^{\,\sharp }_{LC} \ell _{\textit{hom}\hspace{0.2pt}}$, where $\text {d}^{\,\sharp }_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}$ denotes the push forward of the measure $\text {d}_{LC} \ell _{\textit{hom}\hspace{0.2pt}}$; then, by the change of variable formula for the push-forward, see for example [1], we get

$$\begin{aligned} \begin{aligned} \int _{L_0}^{\ell _{ \textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}(l) \, \text {d}l_{|R} = \int _{L_0}^{\ell _{\textit{hom}\hspace{0.2pt}}(t)} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}(l) \, \text {d}^{\,\sharp }_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}= \int _{0}^{t} G^{\hspace{0pt}{ \textit{c}}}_{\hspace{0pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}( \ell _{ \textit{hom}\hspace{0.2pt}}(\tau )) \, \text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}. \end{aligned} \end{aligned}$$

Note that both $\textrm{d}l_{|R}$ and $\text {d}^\sharp _{LC} \ell _{ \textit{hom}\hspace{0.2pt}}$ are Borel measure which do not contain atoms. Thus, it is enough to check that they coincide on closed intervals $[l_{\tiny \text {1}}, l_{\tiny \text {2}}] \subset \textrm{co} \, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}([0,T))$.

To evaluate the push-forward measure of $[l_{\tiny \text {1}}, l_{\tiny \text {2}}]$ we should compute the $(\text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}})$-measure of the set $(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ([l_{\tiny \text {1}}, l_{\tiny \text {2}}])$. If $(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}] ) = \emptyset$ the measure vanishes. Otherwise, by monotonicity of $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ it turns out that $(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}] )$ is again an interval; more precisely, it is of the form $[t_{\tiny \text {1}}, t_{\tiny \text {2}})$ or $[t_{\tiny \text {1}}, t_{\tiny \text {2}}]$, where $t_{\tiny \text {1}}= \min \{ (\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}]) \}$ and $t_{\tiny \text {2}}= \sup \{ (\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ([l_{\tiny \text {1}}, l_{\tiny \text {2}}]) \}$. Note that the point $t_{\tiny \text {1}}$ is always included by the right continuity of $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, moreover, the $\text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}$-measure of $[t_{\tiny \text {1}}, t_{\tiny \text {2}})$ or $[t_{\tiny \text {1}},t_{\tiny \text {2}}]$ actually coincides with the $(\text {d}_{LC} \ell _{\textit{hom}\hspace{0.2pt}})$-measure of the open interval $(t_{\tiny \text {1}}, t_{\tiny \text {2}})$ since $\text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}= {\dot{\ell }}_{ \textit{hom}\hspace{0.2pt}}\text {d}\tau + \text {d}_{C} \ell _{ \textit{hom}\hspace{0.2pt}}$ does not contain atoms. Then

$$\begin{aligned} \begin{aligned} \text {d}_{LC} \ell _{ \textit{hom}\hspace{0.2pt}}(t_{\tiny \text{1 }}, t_{\tiny \text{2 }})&= \text {d} \ell _{ \textit{hom}\hspace{0.2pt}}(t_{\tiny \text{1 }}, t_{\tiny \text{2 }}) - \text {d}_{J} \ell _{ \textit{hom}\hspace{0.2pt}}(t_{\tiny \text{1 }},t_{\tiny \text{2 }}) \\ &= \ell ^-_{ \textit{hom}\hspace{0.2pt}}(t_{\tiny \text{2 }}) - \ell _{ \textit{hom}\hspace{0.2pt}}(t_{\tiny \text{1 }}) - \sum _{ \tau \,\in \, J_{ \textit{hom}\hspace{0.2pt}}\cap \, (t_1, t_2)} \llbracket \ell _{ \textit{hom}\hspace{0.2pt}}(\tau ) \rrbracket . \end{aligned} \end{aligned}$$

(31)

Let us turn to the measure $\textrm{d}l_{|R}$. If $(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}] ) = \emptyset$ then $[l_{\tiny \text {1}}, l_{\tiny \text {2}}] \subset U$ and thus its $\textrm{d}l_{|R}$-measure vanishes. Otherwise, by (30) we can write

$$\begin{aligned} \textrm{d}l_{|R} [ l_{\tiny \text {1}}, l_{\tiny \text {2}}] = ( l_{\tiny \text {2}}- l_{\tiny \text {1}}) - \textrm{d}l_{|U} [ l_{\tiny \text {1}}, l_{\tiny \text {2}}] = (l_{\tiny \text {2}}- l_{\tiny \text {1}}) - \sum _{\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} | U_\tau \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | . \end{aligned}$$

Let us study the measure $| U_\tau \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] |$ as a function of $\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$. Note that $\tau >0$ since $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is right continuous.

If $\tau < t_{\tiny \text {1}}$ then $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) < l_{\tiny \text {1}}$ and thus $| U_\tau \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = 0$. If $\tau > t_{\tiny \text {2}}$ then $\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( \tau ) \geqq l_2$ (since $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) > l_2$ for $t_{\tiny \text {2}}< t < \tau$) and thus $| U_{\tau } \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = 0$.

If $\tau = t_{\tiny \text {1}}$ then $l_1 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) \leqq l_{\tiny \text {2}}$ while $\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) \leqq l_1$ (since $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) < l_1$ for any $t< t_{\tiny \text {1}}$). It follows that $| U_{t_1} \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) - l_{\tiny \text {1}}$. Hence, we can write

$$\begin{aligned} - l_{\tiny \text {1}}- | U_{t_1} \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = - \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) . \end{aligned}$$

If $\tau \in (t_{\tiny \text {1}}, t_{\tiny \text {2}})$ then $l_1 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) \leqq l_{\tiny \text {2}}$ while $\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) \geqq l_{\tiny \text {1}}$ (since $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) \geqq l_1$ for $t_{\tiny \text {1}}\leqq t< \tau$).

Hence $[ \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) ] \subset [ l_1, l_2 ]$ and $| U_\tau \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = \llbracket \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) \rrbracket$.

If $\tau = t_{\tiny \text {2}}$ and $t_2 \not \in (\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}] )$ then $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}}) > l_2$ while $l_1 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^- (t_2) \leqq l_2$ (since $l_1 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^- (t) \leqq l_2$ for every $t \in [t_1, t_{\tiny \text {2}})$). Then $| U_{t_2} \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = l_{\tiny \text {2}}- \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}})$ and we can write

$$\begin{aligned} l_{\tiny \text {2}}- | U_{t_2} \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}}) . \end{aligned}$$

If $\tau = t_{\tiny \text {2}}$ and $t_2 \in (\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})^{-1} ( [l_{\tiny \text {1}}, l_{\tiny \text {2}}] )$ then $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_2) = l_2$; indeed, assume by contradiction that $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_2) < l_2 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(T)$ then $t_{\tiny \text {2}}< T$ and (by right continuity) $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) < l_2$ in a sufficiently small right neighborhood of $t_{\tiny \text {2}}$. Since $l_1 \leqq \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^- (t_2) \leqq l_2$ (as above) we get again

$$\begin{aligned} l_{\tiny \text {2}}- | U_{t_2} \cap [l_{\tiny \text {1}}, l_{\tiny \text {2}}] | = \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}}) . \end{aligned}$$

In conclusion

$$\begin{aligned} \textrm{d}l_{|R} [ l_{\tiny \text {1}}, l_{\tiny \text {2}}] = \ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}}) - \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {1}}) - \hspace{-12pt} \sum _{\tau \in J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap (t_1, t_2) } \llbracket \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) \rrbracket . \end{aligned}$$

(32)

By (32) and (31) the measures $\textrm{d}l_{|R}$ and $\text {d}^\sharp _{LC} \ell _{ \textit{hom}\hspace{0.2pt}}$ do coincide. $\square$

2.7.1 Toughening and Micro-instabilities

In this section we study the toughening effect resulting from our convergence result. As we will see toughening turns out to be the macroscopic effect of microscopic instabilities in the evolutions $\ell _n$.

Let $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ (1-\lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$ be the homogenized (or averaged) toughness, that is the weak$^*$ limit of $G^{\hspace{-2.5pt}{ \textit{c}}}_n$.

For simplicity, let us consider a time interval $[t_a, t_b]$ such that $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_a) < \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_b)$ and $J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}) \cap [t_a , t_b] = \emptyset$. We will show that

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( l ) \ \text {for every }l\in [\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_a) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_b)]. \end{aligned}$$

Let us rewrite the energy identity for $\ell _n$ in the form

$$\begin{aligned} \begin{aligned} E_n ( t , \ell _n(t)) + \int _{L_0}^{\ell _n(t)} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \, \text {d}l = \int _0^t \partial _t F_n ( \tau , \ell _n (\tau )) \, \text {d}\tau + \text {d}_{J} F_n ( [ 0, t] ) , \end{aligned} \end{aligned}$$

(33)

where we used the shorthand notation

$$\begin{aligned} \begin{aligned} \text {d}_{J} F_n ( [ 0, t] ) = \sum _{\tau \in J(\ell _n) \cap [0,t]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket . \end{aligned} \end{aligned}$$

Thus, for every $t_a \leqq t_{\tiny \text {1}}< t_{\tiny \text {2}}\leqq t_b$ we can write

$$\begin{aligned} E_n ( t_{\tiny \text {2}}, \ell _n(t_{\tiny \text {2}})) - E_n ( t_{\tiny \text {1}}, \ell _n(t_{\tiny \text {1}}))&- \int _{t_1}^{t_2} \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau \nonumber \\&= - \int _{\ell _n (t_1)}^{\ell _n(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \, \textrm{d}l + d_{J} F_n ( (t_{\tiny \text {1}}, t_{\tiny \text {2}}] ) . \end{aligned}$$

(34)

Since $\ell _n \rightarrow \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ pointwise in [0, T) by Proposition 7 we get

$$\begin{aligned} E_n ( t_i , \ell _n(t_i) ) \rightarrow E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t_i , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_i) ) \quad \text {for }i=1,2. \end{aligned}$$

Remember that $\partial _t F_n ( \tau , l ) = 2 f (\tau ) \dot{f} (\tau ) \, \mathcal {E}_n ( l)$, hence $\partial _t F_n ( \tau , \ell _n (\tau ) ) \rightarrow \partial _t F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau ) )$ and by dominated convergence we get

$$\begin{aligned} \int _{t_1}^{t_2} \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau \rightarrow \int _{t_1}^{t_2} \partial _t F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \, \textrm{d}\tau . \end{aligned}$$

In summary, the left hand side of (34) converges to

$$\begin{aligned} E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t_{\tiny \text {2}}, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_{\tiny \text {2}})) - E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t_{\tiny \text {1}}, \ell _n(t_{\tiny \text {1}}))&- \int _{t_1}^{t_2} \partial _t F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( \tau , \ell _n (\tau )) \, \textrm{d}\tau \nonumber \\&= - \int _{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_1)}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l , \end{aligned}$$

where the identity follows from Corollary 17 and from the assumption $J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}) \cap [t_a , t_b] = \emptyset$.

As a consequence the right hand side of (34) converges, that is,

$$\begin{aligned} \begin{aligned} \int _{\ell _n (t_1)}^{\ell _n(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \, \text {d}l + \text {d}_{J} F_n ( (t_{\tiny \text{1 }}, t_{\tiny \text{2 }}] ) \rightarrow \int _{\ell _{ \textit{hom}\hspace{0.2pt}}(t_1)}^{\ell _{ \textit{hom}\hspace{0.2pt}}(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}(l) \, \text {d}l . \end{aligned} \end{aligned}$$

As $G^{\hspace{-2.5pt}{ \textit{c}}}_n {\mathop {\rightharpoonup }\limits ^{*}}G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ (1-\lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$,

$$\begin{aligned} \int _{\ell _n (t_{\tiny \text {1}})}^{\ell _n(t_{\tiny \text {2}})} G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \, \textrm{d}l \rightarrow \int _{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_1)}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \, \textrm{d}l . \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} \text {d}_{J} F_n ( (t_{\tiny \text{1 }}, t_{\tiny \text{2 }}] ) \rightarrow \int _{\ell _{ \textit{hom}\hspace{0.2pt}}(t_1)}^{\ell _{ \textit{hom}\hspace{0.2pt}}(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_{ \textit{hom}\hspace{0.2pt}}(l) - G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}(l) \, \text {d}l . \end{aligned} \end{aligned}$$

Since $\text {d}_{J} F_n ( (t_{\tiny \text{1 }}, t_{\tiny \text{2 }}] ) \leqq 0$ and it follows that

$$\begin{aligned} \int _{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_1)}^{\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_2)} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) - G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) \, \textrm{d}l \leqq 0 . \end{aligned}$$

By the arbitrariness of $t_{\tiny \text {1}}$ and $t_{\tiny \text {2}}$ it follows that $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( l )$ almost everywhere in $[\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_a) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_a)]$. Since $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is constant and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ is upper semi-continuous the inequality actually holds everywhere in $[\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_a) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t_b)]$.

2.8 An Explicit Computation of the Effective Toughness

In this section we consider a specific case which highlights some peculiar features of the effective toughness, and in particular its dependence on the data. Besides the hypotheses of the previous section, we assume that $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-4pt} A},{\tiny \text {2}}}$. No further assumptions are made on the toughness, thus it may occur $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\ne G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$ or $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}= G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}$. These assumptions include for instance the case in which material A is anisotropic and material B is obtained by a $\frac{\pi }{2}$ rotation of material A.

For convenience, let us recall the results of Proposition 7 and a few notations. The homogenized stiffness matrix is given by

$$\begin{aligned} \, \varvec{C}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} = \left( \begin{array}{cc} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} \end{array} \right) , \end{aligned}$$

where $1 / \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {1}}} = \lambda / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + (1-\lambda ) / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ and $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}$. The (condensed) elastic energy $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is of class $C^1 (0,L)$ and the energy release is denoted by $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. We will prove that

$$\begin{aligned} \begin{aligned} G^{{ \textit{c}}}_{ \textit{ef}\hspace{0.15pt} \textit{f}\hspace{1pt}}= \lambda \max \left\{ G^{{ \textit{c}}}_{ A}\, , \, G^{{ \textit{c}}}_{ B}\, \frac{\mu _{{ B},{\tiny \text{1 }}}}{\mu _{{ A},{\tiny \text{1 }}}} \right\} + ( 1 - \lambda ) \max \left\{ G^{{ \textit{c}}}_{ A}\frac{\mu _{{ A},{\tiny \text{1 }}}}{\mu _{{ B},{\tiny \text{1 }}}} \, , \, G^{{ \textit{c}}}_{ B}\right\} . \end{aligned} \end{aligned}$$

(35)

In particular, the effective toughness is constant and depends also on the elastic contrast $\mu _{{\hspace{-4pt} A},{\tiny \text {1}}} / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$. More precisely, we will prove that $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ takes the above value where $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\ne 0$, which is enough for our purpose in the light of Remark 5.

Example 1

Assume that $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} < \mu _{{\hspace{-4pt} A},{\tiny \text {1}}}$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}$. Then

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}= \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ ( 1 - \lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\frac{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} > \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ ( 1 - \lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}= G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}= \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} . \end{aligned}$$

Next theorem states the main convergence result.

Theorem 18

Let $\ell _n$ be the quasi-static evolutions given by Proposition 2. Then $\ell _n {\mathop {\rightharpoonup }\limits ^{*}}\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ in BV(0, T), where $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is the unique non-decreasing, right continuous function which satisfies the initial condition $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(0) = L_0$ and Griffith’s criterion in the following form:

• (i) $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ for every time $t \in [0,T)$;

• (ii) if $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ then $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is differentiable in t and $\dot{\ell }_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) =0$;

• (iii) if $t \in J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$ for every $l \in [\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) )$.

Note that in the above statement it is not necessary to extract any subsequence of $\ell _n$. As the effective toughness is constant, the effective dissipated energy boils down to $D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l) = G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( l - L_0)$; by Corollary 5 we get the energy identity for $F (t, u) = E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, u) + D_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$, stated hereafter.

Corollary 19

For every $t \in [0,T)$, the following energy identity holds:

$$\begin{aligned} F ( t , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)) = \int _0^t \partial _t F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \, \textrm{d}\tau + \hspace{-12pt} \sum _{\tau \, \in \, J_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\cap [0,t]} \llbracket F ( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \rrbracket . \end{aligned}$$

(36)

Remark 6

Note that the limit energy $F (t, \cdot )$ does not coincide with the $\Gamma$-limit of the energies $F_n (t,\cdot ) = E_n (t, \cdot ) + D_n (\cdot )$, which is given by

$$\begin{aligned} F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t,l) = E_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, l) + G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l - L_0) \ \text { where } \ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \lambda G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}+ (1-\lambda ) G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}. \end{aligned}$$

Before proceeding we state, for the reader’s convenience, the following elementary change of variable, which will be often used in the sequel:

Lemma 20

Let O be an open set in $\mathbb {R}^2$ and ${{\varvec{C}}} \in \mathbb {R}^{2 \times 2}$. Let $\Phi : O \rightarrow {\widehat{O}}$ be a bi-Lipschitz map. For $z \in H^1(O)$ denote $\hat{z} = ( z \circ \Phi ^{-1}) \in H^1({\widehat{O}})$, then

$$\begin{aligned} \int _O \nabla z \, {{\varvec{C}}} \, \nabla z^T \textrm{d}x = \int _{{\widehat{O}}} \nabla \hat{z} \, \widehat{{\varvec{C}}} \nabla \hat{z}^T d\hat{x} \quad \text {where} \quad \widehat{{\varvec{C}}} = (D\Phi ) {{\varvec{C}}} (D\Phi )^{T} ( \textrm{det}D\Phi ) ^{-1} . \end{aligned}$$

(37)

In particular, if $\Phi ( x_{\tiny \text {1}}, x_{\tiny \text {2}}) = ( \alpha x_{\tiny \text {1}}, \beta x_{\tiny \text {2}})$ and $\, \varvec{C}= \mu _{\tiny \text {1}}\hat{e}_{\tiny \text {1}}\otimes \hat{e}_{\tiny \text {1}}+ \mu _{\tiny \text {2}}\hat{e}_{\tiny \text {2}}\otimes \hat{e}_{\tiny \text {2}}$ then

$$\begin{aligned} \widehat{{\varvec{C}}} = \left( \begin{array}{cc} \mu _{{\tiny \text {1}}} \alpha / \beta & 0 \\ 0 & \mu _{{\tiny \text {2}}} \beta / \alpha \end{array} \right) . \end{aligned}$$

Proof of Theorem 18

The Theorem follows from

$$\begin{aligned} \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{\displaystyle \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_{n}}{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} (l) } = \lambda \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\, \frac{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} \right\} + ( 1 - \lambda ) \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\frac{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} \, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\right\} , \end{aligned}$$

(38)

for $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$, after employing Remark 3 and Remark 5.

I. Let $\alpha = \mu _{\,{\hspace{-3.5pt} B},{\tiny \text {1}}} / \mu _{\,{\hspace{-4pt} A},{\tiny \text {1}}}$. For $S = \lambda L \alpha + (1-\lambda )L$ let ${\widehat{\Omega }} = (0, S) \times (-H,H)$. We consider a bi-Lipschitz piecewise affine map $\Phi _n : \Omega \rightarrow {\widehat{\Omega }}$ of the form $\Phi _n ( x_{\tiny \text {1}}, x_2 ) = (\phi _n ( x_{\tiny \text {1}}) , x_2 )$ where

$$\begin{aligned} \phi _n (x_1) = \int _0^{x_1} \alpha _n \, \textrm{d}s \quad \text {and } \quad \alpha _n = {\left\{ \begin{array}{ll} \alpha & \text {in }K_{n,\,{\hspace{-4pt} A}} \\ 1 & \text {in }K_{n,\,{\hspace{-3.5pt} B}}. \end{array}\right. } \end{aligned}$$

We employ the notation ${\widehat{\Omega }}_{n,\,{\hspace{-4pt} A}}$ to denote the set $\Phi _n ( \Omega _{n,\,{\hspace{-4pt} A}} )$ and similarly for all the sets introduced in §2.1. We denote $\hat{g}_n = g \circ \Phi _n^{-1}$ and consider the spaces

$$\begin{aligned} {\widehat{\mathcal {U}}}_{s} = \{ \hat{u} \in H^1({\widehat{\Omega }} {\setminus } K_s) : \hat{u} = \hat{g}_n \text { in } \partial _D {\widehat{\Omega }} \} \quad \text {for }s \in (0,S]. \end{aligned}$$

Let $s = \phi _n(l)$. For $u \in \mathcal {U}_{{\hspace{0.5pt}l}}$ and $\hat{u} = u \circ \Phi _n^{-1} \in {\widehat{\mathcal {U}}}_{s}$ we have and

$$\begin{aligned} \mathcal {W}_{l,n} (u ) = {\widehat{\mathcal {W}}}_{s,n} ( \hat{u} ) = \tfrac{1}{2} \int _{{\widehat{\Omega }} {\setminus } \widehat{K}_{s} } \nabla \hat{u} \, \widehat{{\varvec{C}}}_n \nabla \hat{u}^T \, \textrm{d}y . \end{aligned}$$

Under the assumption $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-4pt} A},{\tiny \text {2}}}$ and with the above choice of $\alpha = \mu _{\,{\hspace{-3.5pt} B},{\tiny \text {1}}} / \mu _{\,{\hspace{-4pt} A},{\tiny \text {1}}}$, it turns out that $\widehat{{\varvec{C}}}_n$ is constant in ${\widehat{\Omega }}$; indeed we have

$$\begin{aligned} \widehat{{\varvec{C}}}_n = \left( \begin{array}{cc} \alpha \, \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} / \alpha \end{array} \right) = \left( \begin{array}{cc} \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} \end{array} \right) \ \text {in } {\widehat{\Omega }}_{A,n} , \quad \widehat{{\varvec{C}}}_n = \left( \begin{array}{cc} \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} \end{array} \right) \ \text {in }{\widehat{\Omega }}_{B,n}. \end{aligned}$$

Note also that $\widehat{{\varvec{C}}}_n$ is actually independent of n; therefore in the sequel we will skip the dependence on n in the notation. The reduced energy ${\widehat{\mathcal {E}}}$ is then of class $C^1 (0,S)$; in particular the energy release ${\widehat{\hspace{0.5pt}\mathcal {G}}}$ is continuous in (0, S). Clearly, by the change of variable,

$$\begin{aligned} \widehat{\mathcal {E}} (\phi _n (l) ) = \mathcal {E}_n (l) \quad \text {and thus} \quad \widehat{\hspace{0.5pt}\mathcal {G}} (\phi _n (l)) \,\phi _n' (l) = \hspace{0.5pt}\mathcal {G}_n ( l ) \ \text {for } l \in (0,L) {\setminus } \Lambda _n. \end{aligned}$$

(39)

II. Let $\alpha _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} = \alpha \lambda + ( 1- \lambda )$. Let $\Phi : \Omega \rightarrow {\widehat{\Omega }}$ be the linear map $\Phi ( x_{\tiny \text {1}}, x_2 ) = ( \phi (x_{\tiny \text {1}}) , x_2 )$ for $\phi (x_1) = \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}x_1$. With this change of variable, for $s = \phi (l) = \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}l$, we write

$$\begin{aligned} \mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} (u) = \widehat{\mathcal {W}}_{s,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} ( \hat{u} ) = \tfrac{1}{2} \int _{{\widehat{\Omega }} {\setminus } {\widehat{K}}_s} \nabla \hat{u} \, \widehat{\textbf{C}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\, \nabla \hat{u}^T \, \textrm{d}y , \end{aligned}$$

where

$$\begin{aligned} \widehat{{\varvec{C}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \left( \begin{array}{cc} \alpha _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} / \alpha _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \end{array} \right) . \end{aligned}$$

As expected, $\widehat{\textbf{C}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \widehat{\textbf{C}}$; indeed,

$$\begin{aligned} \frac{1}{\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}}}= & \frac{\lambda }{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} + \frac{1-\lambda }{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} = \frac{\lambda \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} + (1-\lambda ) \mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} , \quad \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \frac{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} \lambda + ( 1- \lambda ) \\ & = \frac{\lambda \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} + (1-\lambda ) \mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} , \end{aligned}$$

and thus $\alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\, \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \mu _{{\hspace{-3.5pt} B}, {\tiny \text {1}}}$, moreover, recalling that $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-4pt} A},{\tiny \text {2}}}$,

$$\begin{aligned} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} , \quad \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \frac{\mu _{{\hspace{-4pt} A},{\tiny \text {2}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}} \lambda + ( 1- \lambda ) = \frac{\lambda \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}} , \end{aligned}$$

and then $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} / \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \mu _{{\hspace{-3.5pt} B}, {\tiny \text {2}}}$. In conclusion,

$$\begin{aligned} {\widehat{\mathcal {E}}} ( \phi (l) ) = \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \quad \text {and thus} \quad {\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi ( l ) ) \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ \text {for } l \in (0,L). \end{aligned}$$

(40)

III. By (39) we can write

$$\begin{aligned} \frac{\hspace{0.5pt}\mathcal {G}_{n} (l) }{G^{\hspace{-2.5pt}{ \textit{c}}}_{n} (l) } = \frac{{\widehat{\hspace{0.5pt}\mathcal {G}}} (\phi _n (l) ) \, \phi '_n (l) }{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}(l)} . \end{aligned}$$

Note that $\phi _n \rightarrow \phi$ uniformly in [0, L]. Then, being ${\widehat{\hspace{0.5pt}\mathcal {G}}}$ continuous, ${\widehat{\hspace{0.5pt}\mathcal {G}}} \circ \phi _n \rightarrow {\widehat{\hspace{0.5pt}\mathcal {G}}} \circ \phi$ locally uniformly in (0, L). As a consequence

$$\begin{aligned} \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_{n}}{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} (l) = {\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi (l) ) \, \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\phi '_n }{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} (l) . \end{aligned}$$

(41)

Writing, explicitly,

$$\begin{aligned} \frac{\, \phi '_n (l) }{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}(l)} = {\left\{ \begin{array}{ll} \alpha / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}& \text { in }K_{n,{\hspace{-4pt} A}}, \\ 1 / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}& \text { in }K_{n,{\hspace{-3.5pt} B}}, \end{array}\right. } \end{aligned}$$

we get $\Gamma \text {-}\lim _{n \rightarrow \infty } {\displaystyle \frac{\phi '_n}{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} } (l) = \min \{ \alpha / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, 1 / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\}$. If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) \ne 0$, then ${\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi (l) ) \ne 0$, and by (40) and (41),

$$\begin{aligned} G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)&= \frac{\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) }{\displaystyle \Gamma \text {-}\hspace{-4pt}\lim _{n \rightarrow \infty } \frac{\hspace{0.5pt}\mathcal {G}_{n}}{G^{\hspace{-2.5pt}{ \textit{c}}}_{n}} (l) } = \frac{ \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}{\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi (l) ) }{ {\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi (l) ) \min \{ \alpha / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}, 1 / G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} }\\ &= \big ( \alpha \lambda + ( 1- \lambda ) \big ) \max \{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}/ \alpha , G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\} \\&= \lambda \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\, \frac{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}} \right\} + ( 1 - \lambda ) \max \left\{ G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-4pt} A}\frac{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}}}{\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} \, , \, G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} B}\right\} . \end{aligned}$$

The proof of (38) is concluded. $\square$

Remark 7

In general $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is neither the pointwise nor the $\Gamma$-limit of $\hspace{0.5pt}\mathcal {G}_n$. Indeed, by (39) and (40) we have

$$\begin{aligned} \widehat{\hspace{0.5pt}\mathcal {G}} (\phi _n (l)) \,\phi _n' (l) = \hspace{0.5pt}\mathcal {G}_n ( l ) \ \text { and } \ {\widehat{\hspace{0.5pt}\mathcal {G}}} ( \phi ( l ) ) \alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) . \end{aligned}$$

We know that $\widehat{\hspace{0.5pt}\mathcal {G}} \circ \phi _n \rightarrow {\widehat{\hspace{0.5pt}\mathcal {G}}} \circ \phi$ locally uniformly but the pointwise limit of $\phi '_n$ does not exists. Moreover, $\Gamma \text {-}\lim _{n \rightarrow \infty } \phi '_n (l) = \min \{ \alpha , 1 \}$, while $\alpha _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \lambda \alpha + ( 1- \lambda ) = \min \{ \alpha , 1 \}$ if and only if $\alpha =1$; the latter case is trivial, since $\mu _{{\hspace{-3.5pt} B}, {\tiny \text {1}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}}$ and $\mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-4pt} A},{\tiny \text {2}}}$ imply that the stiffness matrix ${{\varvec{C}}}_n$ is constant.

3 Horizontal Layers with Horizontal Crack

3.1 Energy and Energy Release

As in the previous section we assume that the uncracked reference configuration is the set $\Omega = (0,L) \times (-H,H)$ and that the crack set is of the form $K_l = [0, l] \times \{ 0 \}$ for $l \in (0,L]$. On the contrary, here we assume that $\Omega$ is made periodic horizontal layers of thickness $h_n = H/n$ for $n \in 2 \mathbb {N}$ (the fact that n is even implies that $K_l$ is contained in the interface between two layers). For $\lambda \in (0,1)$ each layer is itself made of a horizontal layer of material A, with thickness $\lambda h_n$, and an horizontal layer of material B, with thickness $(1-\lambda ) h_n$ (see Fig. 2). In this geometry crack may actually propagate along the interface or penetrate into the layer [12]. The convergence result presented in the sequel holds only in the former case, since the latter is still out of reach in our analysis.

Fig. 2

An example of geometry in the case of an interfacial crack with horizontal layers

For functional spaces, boundary condition, energies and energy release we employ the same assumptions and notation of the previous section, apart from toughness which here is simply denoted by $G^{\hspace{-2.5pt}{ \textit{c}}}$, since it is independent of n.

Following the proof of Lemma 1 it not difficult to show

Lemma 21

$\mathcal {E}_n : [0,L] \rightarrow \mathbb {R}$ is decreasing and continuous. Moreover, it is of class $C^1 (0,L)$ and

$$\begin{aligned} \mathcal {E}'_n ( l) = \int _{\Omega {\setminus } K_l} \nabla u_{l,n} \varvec{C}_n \varvec{E} \nabla u_{l,n}^{T} \, \phi ' \textrm{d}x \quad \text {for} \quad \varvec{E} = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) , \end{aligned}$$

(42)

where $\phi \in W^{1,\infty } (0,L)$ with $\textrm{supp} ( \phi ) \subset (0,L)$, $0 \leqq \phi \leqq 1$ and $\phi (l) =1$.

Remark 8

Comparing with Lemma 1 note that here $\phi$ is independent of n, in particular its support does not shrink with n. Finally, note that $\mathcal {E}_n$ is of class $C^1$ in the whole (0, L).

3.2 Quasi-Static Evolution by Griffith’s Criterion

Following the proofs of Proposition 2 and Corollary 5 we obtain the following results:

Proposition 22

There exists a unique non-decreasing, right continuous function $\ell _n : [0,T) \rightarrow [L_0, L]$ which satisfies the initial condition $\ell _n(0) = L_0$ and Griffith’s criterion, in the following form:

• (i) $G_n (t, \ell _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}$ for every time $t \in [0,T)$;

• (ii) if $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}$ then $\ell _n$ is differentiable in t and ${\dot{\ell }}_n (t) = 0$;

• (iii) if $t \in J ( \ell _n)$ then $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}$ for every $l \in [\ell ^-_n(t) , \ell _n(t) )$.

Corollary 23

If $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}$ then $\ell _n$ is constant in a neighborhood of t.

Corollary 24

For every $t \in [0,T)$ the following energy identity holds:

$$\begin{aligned} F_n ( t , \ell _n (t)) = \int _0^t \partial _t F_n ( \tau , \ell _n (\tau )) \, \textrm{d}\tau + \sum _{\tau \in J(\ell _n) \cap [0,t]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket . \end{aligned}$$

(43)

3.3 Homogenization of Griffith’s Criterion

Here the homogenized stiffness matrix is

$$\begin{aligned} \, \varvec{C}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} = \left( \begin{matrix} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} \end{matrix} \right) , \end{aligned}$$

where $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ is the weak* limit of $\mu _{n,{\tiny \text {1}}}$ while $1 / \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} = \lambda / \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + (1-\lambda ) / \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}$ is the weak* limit of $1/\mu _{n,{\tiny \text {2}}}$.

Accordingly, we introduce the homogenized elastic energy $\mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ and the reduced homogenized energy $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$. Arguing as in Lemma 1, it turns out that the energy $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is of class $C^{1} (0,L)$, and hence the energy release $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l) = -\partial _l \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$ is well defined in (0, L). We set $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(L)=0$.

In this setting it turns out that $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}= G^{\hspace{-2.5pt}{ \textit{c}}}$; indeed, we have the following result:

Theorem 25

Let $\ell _n$ be the quasi-static evolutions given by Proposition 22. There exists a subsequence (not relabeled) such that $\ell _n \rightarrow \ell$ pointwise in [0, T). Then $\ell ^+ = \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, where $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is the unique non-decreasing, right continuous function which satisfies the initial condition $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(0) = L_0$ and Griffith’s criterion:

• (i) $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}$ for every time $t \in [0,T)$;

• (ii) if $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}$ then ${\dot{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) = 0$;

• (iii) if $t \in J ( \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}$ for every $l \in [\ell ^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t) )$.

Corollary 26

For every $t \in [0,T)$ the following energy identity holds:

$$\begin{aligned} F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t)) = \int _0^t \partial _t F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \, \textrm{d}\tau + \sum _{\tau \in J(\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}) \cap [0,t]} \llbracket F_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( \tau , \ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(\tau )) \rrbracket . \end{aligned}$$

(44)

The proof of Theorem 25 is a consequence of Corollary 15 and Remark 5, together with the following result on the convergence of the energy release:

Theorem 27

$\hspace{0.5pt}\mathcal {G}_n$ converge to $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ locally uniformly in (0, L).

The proof of this theorem is contained in the next subsection.

3.4 Convergence of the Energy Release

Lemma 28

$\hspace{0.5pt}\mathcal {G}_n$ is uniformly bounded in $C^{\,0, 1/2}( [ L',L'' ])$ for every $0< L'< L'' < L$.

Proof

By minimality $\mathcal {W}_{l,n} ( u_{l,n} ) \leqq \mathcal {W}_{l,n} (g) \leqq C$ where C is independent of n. Hence, by uniform coercivity, $\Vert \nabla u_{l,n} \Vert _{L^2} \leqq C$ for every $l \in [0,L]$.

Now, following [20] we prove that there exists $C>0$ such that, for every $0 \leqq l_{\tiny \text {1}}< l_{\tiny \text {2}}\leqq L$ and every $n \in 2 \mathbb {N}$, it holds that

$$\begin{aligned} \Vert \nabla u_{l_1,n} - \nabla u_{l_2,n} \Vert _{L^2} \leqq C | \mathcal {E}_n ( l_{\tiny \text {1}}) - \mathcal {E}_n (l_{\tiny \text {2}}) | ^{1/2} . \end{aligned}$$

(45)

By the variational formulation we have

$$\begin{aligned} & \int _{ \Omega {\setminus } K_{l_1}} \nabla {u}_{l_1,n} \, \varvec{C}_n \nabla v_{{\tiny \text {1}}}^T \, \textrm{d}y = 0 \quad \text {for every } v_{{\tiny \text {1}}} \in \mathcal {V}_{{\hspace{0.5pt}l}_1}, \\ & \int _{ \Omega {\setminus } K_{l_2}} \nabla {u}_{l_2,n} \, \varvec{C}_n \nabla v_{{\tiny \text {2}}} ^T \, \textrm{d}y = 0 \quad \text {for every } v_{{\tiny \text {2}}} \in \mathcal {V}_{{\hspace{0.5pt}l}_2}. \end{aligned}$$

Since $( {u}_{l_1,n} - {u}_{l_2,n} ) \in \mathcal {V}_{{\hspace{0.5pt}l}_2}$, we get

$$\begin{aligned} \int _{ \Omega {\setminus } K_{l_2}} \nabla {u}_{l_2,n} \, \varvec{C}_n \nabla ( {u}_{l_1,n} - {u}_{l_2,n} )^T \, \textrm{d}y = 0 . \quad \end{aligned}$$

Then, by monotonicity and uniform coercivity of the energy, we get

$$\begin{aligned} | \mathcal {E}_n ( l_{\tiny \text {1}}) - \mathcal {E}_n (l_{\tiny \text {2}}) |&= \mathcal {E}_{n} ( l_{\tiny \text {1}}) - \mathcal {E}_n (l_{\tiny \text {2}})\\ &= \tfrac{1}{2} \int _{ \Omega } \nabla {u}_{l_1,n} \, \varvec{C}_n \nabla {u}_{l_1,n}^T - \nabla {u}_{l_2,n} \, \varvec{C}_n \nabla {u}_{l_2,n}^T \, \textrm{d}y \\&= \tfrac{1}{2} \int _{ \Omega } \nabla ( {u}_{l_1,n} + {u}_{l_2,n}) \, \varvec{C}_n \nabla ( {u}_{l_1,n} - {u}_{l_2,n})^T \, \textrm{d}y \\&= \tfrac{1}{2} \int _{ \Omega } \nabla {u}_{l_1,n} \, \varvec{C}_n \nabla ( {u}_{l_1,n} - {u}_{l_2,n})^T \, \textrm{d}y\\&= \tfrac{1}{2} \int _{ \Omega } \nabla ( {u}_{l_1,n} - {u}_{l_2,n}) \, \varvec{C}_n \nabla ( {u}_{l_1,n} - {u}_{l_2,n})^T \, \textrm{d}y\\&\geqq C \Vert \nabla {u}_{l_1,n} - \nabla {u}_{l_2,n} \Vert _{L^2}^2 . \end{aligned}$$

Next, we show that given $0< L'< L''< L$ there exists $C >0$ s.t.

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_n (l) \leqq C \quad \text {for every }l \in [L',L'']\text { and every } n \in 2 \mathbb {N}. \end{aligned}$$

Let $\phi \in C_c^{\infty } (0,L)$ with $0 \leqq \phi \leqq 1$ and $\phi =1$ in $[L',L'']$ such that

$$\begin{aligned} \mathcal {E}'_n (l) = \int _{ \Omega } \nabla u_{l,n} \, \varvec{C}_n \varvec{E} \, \nabla {u}_{l,n}^T \, \phi ' \, \textrm{d}y \quad \text {for} \quad \varvec{E} = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) . \end{aligned}$$

Hence $\hspace{0.5pt}\mathcal {G}_n (l) \leqq C$ (note that C depends on $\phi '$ and thus on $L'$ and $L''$). It follows that the energies $\mathcal {E}_n$ are uniformly Lipischitz continuous in $[L',L'']$ and thus, by (45),

$$\begin{aligned} \Vert \nabla u_{l_1,n} - \nabla u_{l_2,n} \Vert _{L^2} \leqq C | l_1 - l_2 | ^{1/2} . \end{aligned}$$

As a consequence, using the integral representation of $\hspace{0.5pt}\mathcal {G}_n = - \mathcal {E}'_n$ we get

$$\begin{aligned} | \hspace{0.5pt}\mathcal {G}_n ( l_{\tiny \text {1}}) - \hspace{0.5pt}\mathcal {G}_n ( l_{\tiny \text {2}}) | \leqq C | l_{\tiny \text {1}}- l_{\tiny \text {2}}|^{1/2} , \end{aligned}$$

which concludes the proof. $\square$

By Ascoli-Arzelà Theorem, with the aid of a diagonal argument, Lemma 28 implies

Corollary 29

There exists a subsequence (non relabeled) and a limit $\hspace{0.5pt}\mathcal {G}$ such that $\hspace{0.5pt}\mathcal {G}_n \rightarrow \hspace{0.5pt}\mathcal {G}$ locally uniformly in (0, L).

We claim that $\hspace{0.5pt}\mathcal {G}=\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$. To this end, we will need a couple of lemmas on the local convergence of the energy.

Lemma 30

For $0< a< b < L$, let $\Omega _{(a,b)} = (a,b) \times (- H , H)$. Then

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l } \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}_{l,n}^T \,\textrm{d}x \ \rightarrow \ \int _{\Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \,\textrm{d}x . \end{aligned}$$

Proof

For $\varepsilon >0$ let $\theta _\varepsilon \in C^\infty _c (0,L)$ with $0 \leqq \theta _\varepsilon \leqq 1$, $\theta _\varepsilon =1$ in (a, b), and such that

$$\begin{aligned} \int _{ \Omega {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \theta _\varepsilon \, \textrm{d}x \leqq \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

Then, by the properties of $\theta _\varepsilon$ and since ${u}_{l,n} \theta _\varepsilon \in \mathcal {V}_l$ by the variational formulation we can write

$$\begin{aligned} \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}_{l,n}^T \, \textrm{d}x&\leqq \int _{ \Omega {\setminus } K_l} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}_{l,n}^T \theta _\varepsilon \, \textrm{d}x \\&= \int _{ \Omega {\setminus } K_l}\nabla {u}_{l,n} \, \varvec{C}_n \left( \nabla (u_{l,n} \theta _\varepsilon ) - {u}_{l,n} \nabla \theta _\varepsilon \right) ^T \textrm{d}x \\&= - \int _{ \Omega {\setminus } K_l}\nabla {u}_{l,n} \, \varvec{C}_n \nabla \theta _\varepsilon ^T {u}_{l,n} \, \textrm{d}x . \end{aligned}$$

As $\, \varvec{C}_n \nabla {u}_{l,n}^T \rightharpoonup \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T$ in $L^2 ( \Omega {\setminus } K_l; \mathbb {R}^2)$ and since ${u}_{l,n} \rightarrow {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ in $L^2( \Omega {\setminus } K_l)$ the last term converge to

$$\begin{aligned} - \int _{ \Omega {\setminus } K_l}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}&\, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla \theta _\varepsilon ^T {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \textrm{d}x = \\&= - \int _{ \Omega {\setminus } K_l}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\left( \nabla ( {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \theta _\varepsilon ) - \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \theta _\varepsilon \right) ^T \textrm{d}x \\&= \int _{ \Omega {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \theta _\varepsilon \, \textrm{d}x \\&\leqq \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

Hence

$$\begin{aligned} \limsup _{n \rightarrow \infty } \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,n} \varvec{C}_n \nabla {u}_{l,n}^T \,\textrm{d}x \leqq \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

For $\varepsilon >0$ let $\psi _\varepsilon \in C^\infty _c ( 0, L )$ with $0 \leqq \psi _\varepsilon \leqq 1$, $\psi _\varepsilon = 0$ in $(0,L) {\setminus } (a,b)$, and such that

$$\begin{aligned} \int _{ \Omega {\setminus } K_l}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \psi _\varepsilon \, \textrm{d}x \geqq - \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

Then, arguing as above,

$$\begin{aligned} \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,n} {\varvec{C}}_n \nabla {u}_{l,n}^T \,\textrm{d}x&\geqq \int _{ \Omega {\setminus } K_l} \nabla {u}_{l,n} {\varvec{C}}_n \nabla {u}_{l,n}^T \psi _\varepsilon \, \textrm{d}x, \end{aligned}$$

where the last term converges to

$$\begin{aligned} \int _{ \Omega {\setminus } K_l} \hspace{-5pt} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \psi _\varepsilon \, \textrm{d}x \geqq - \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

Hence

$$\begin{aligned} \liminf _{n \rightarrow \infty } \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,n} {\varvec{C}}_n \nabla {u}_{l,n}^T \,\textrm{d}x \geqq - \varepsilon + \int _{ \Omega _{(a,b)} {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T \, \textrm{d}x . \end{aligned}$$

We conclude by the arbitrariness of $\varepsilon >0$. $\square$

Lemma 31

For $0< a< b < L$, let $\Omega _{(a,b)} = (a,b) \times (- H , H)$. Then

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,n} |^2 \,\textrm{d}x \ \rightarrow \ \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} \, | \partial _{x_1} u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x . \end{aligned}$$

(46)

Proof

We employ a change of variable. Let ${\widehat{\Omega }} = (0, L) \times (- S , S )$ where $S = H \left( \lambda / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} + (1-\lambda ) / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \right)$. Denote by $\Pi _{n,\,{\hspace{-4pt} A}}$ the projection of $\Omega _{n,\,{\hspace{-4pt} A}}$ on the verical axis, that is,

$$\begin{aligned} \Pi _{n,\hspace{1pt}{\hspace{-4pt} A}} = \{ x_2 \in (-H,H) : (x_1, x_2) \in \Omega _{n,\hspace{1pt}{\hspace{-4pt} A}} \text { for }x_1 \in (0,L) \} , \end{aligned}$$

and similarly for $\Pi _{n,\hspace{1pt}{\hspace{-3.5pt} B}}$. Let $\Phi _n : \Omega \rightarrow {\widehat{\Omega }}$ be the bi-Lipschitz piecewise affine map $\Phi _n ( x_{\tiny \text {1}}, x_2 ) = ( x_{\tiny \text {1}}, \phi _n (x_2) )$ where

$$\begin{aligned} \phi _n (x_2) = \int _0^{x_2} \beta _n \, \textrm{d}s \quad \text {and } \quad \beta _n = {\left\{ \begin{array}{ll} 1 / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} & \text {in } \Pi _{n,\hspace{1pt}{\hspace{-4pt} A}}, \\ 1/ \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} & \text {in } \Pi _{n,\hspace{1pt}{\hspace{-3.5pt} B}}. \end{array}\right. } \end{aligned}$$

Denote $\hat{g}_n = g \circ \Phi _n^{-1}$ and

$$\begin{aligned} {\widehat{\mathcal {U}}}_{{\hspace{0.5pt}l},n} = \{ \hat{u} \in H^1 ({\widehat{\Omega }} {\setminus } K_l) : \hat{u} = \hat{g}_n \text { in } \partial _D {\widehat{\Omega }} \} \quad {\widehat{\mathcal {V}}}_l = \{ \hat{v} \in H^1 ({\widehat{\Omega }} {\setminus } K_l) : \hat{v} = 0 \text { in } \partial _D {\widehat{\Omega }} \} . \end{aligned}$$

For $u \in \mathcal {U}_{{\hspace{0.5pt}l}}$ and ${\hat{u}}= ( u \circ \Phi _n^{-1} ) \in {\widehat{\mathcal {U}}}_{{\hspace{0.5pt}l},n}$ we have

$$\begin{aligned} \mathcal {W}_{l,n} (u ) = \widehat{\mathcal {W}}_{l,n} ( {\hat{u}} ) = \tfrac{1}{2} \int _{{\widehat{\Omega }} {\setminus } K_{l} } \nabla {\hat{u}} \, \widehat{{\varvec{C}}}_n \nabla {\hat{u}} ^T \, \textrm{d}y \end{aligned}$$

where (see Lemma 20)

$$\begin{aligned} \widehat{{\varvec{C}}}_n = \left( \begin{matrix} {\hat{\mu }}_{n,{\tiny \text {1}}} & 0 \\ 0 & {\hat{\mu }}_{n,{\tiny \text {2}}} \end{matrix} \right) \quad \text {with} \quad {\hat{\mu }}_{n,{\tiny \text {1}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \quad \text {and} \quad {\hat{\mu }}_{n,{\tiny \text {2}}} = {\left\{ \begin{array}{ll} \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} & \text {in } {\widehat{\Omega }}_{n, \hspace{1pt}{\hspace{-4pt} A}}, \\ \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} & \text {in }{\widehat{\Omega }}_{n,\hspace{1pt}{\hspace{-3.5pt} B}}. \end{array}\right. } \end{aligned}$$

Note that $\hat{\mu }_{n, {\tiny \text {1}}}$ is constant and independent of $n \in 2 \mathbb {N}$. We denote $\hat{\mu }_{{\hspace{-4pt} A},{\tiny \text {2}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ and $\hat{\mu }_{{\hspace{-3.5pt} B},2} = \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}}$. Note that the relative size of the layers ${\widehat{\Omega }}_{n,\hspace{1pt}{\hspace{-4pt} A}}$ and ${\widehat{\Omega }}_{n,\hspace{1pt}{\hspace{-3.5pt} B}}$ is, respectively,

$$\begin{aligned} {\hat{\lambda }} = \frac{ \lambda / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} }{ \lambda / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} + ( 1 - \lambda ) / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} } = \frac{ \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} }{ \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + ( 1 - \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} , \quad ( 1 - {\hat{\lambda }} ) = \frac{ ( 1 - \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} }{ \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + ( 1 - \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} . \end{aligned}$$

Therefore, by Proposition 32, the homogenized stiffness matrix in the rescaled domain ${\widehat{\Omega }}$ is given by

$$\begin{aligned} \widehat{{\varvec{C}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}= \left( \begin{matrix} {\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} & 0 \\ 0 & {\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} \end{matrix} \right) , \end{aligned}$$

where ${\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$, while

$$\begin{aligned} \frac{ 1 }{ {\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} } = \frac{ {\hat{\lambda }} }{ {\hat{\mu }}_{{\hspace{-4pt} A},{\tiny \text {2}}} } + \frac{ 1 - {\hat{\lambda }} }{ {\hat{\mu }}_{{\hspace{-3.5pt} B},{\tiny \text {2}}} } = \frac{ \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} }{ \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + ( 1 - \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} } \left( \frac{\lambda }{ \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} } + \frac{(1-\lambda ) }{ \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} } \right) . \end{aligned}$$

Next, let $\Phi : \Omega \rightarrow {\widehat{\Omega }}$ be the bi-Lipschitz affine map $\Phi ( x_{\tiny \text {1}}, x_2 ) = ( x_{\tiny \text {1}}, \phi (x_2) )$, where

$$\begin{aligned} \phi (x_2) = \beta x_2 \quad \text { for } \quad \beta = \lambda / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} + (1-\lambda ) / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} = \frac{\lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + (1-\lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}}{\mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}} . \end{aligned}$$

Note that $\beta _n {\mathop {\rightharpoonup }\limits ^{*}}\beta$. With this change of variable the stiffness matrix $\, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ becomes $\widehat{{\varvec{C}}}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, indeed (see Lemma 20),

$$\begin{aligned} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + (1-\lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} \quad \text {and} \quad \frac{1}{\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}}} = \frac{\lambda }{ \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} } + \frac{(1-\lambda ) }{ \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} }, \end{aligned}$$

and thus $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} / \beta = \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} = {\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}}$ and

$$\begin{aligned} \frac{1}{ \beta \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} } = \frac{ \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} }{ \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + ( 1 - \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}} } \left( \frac{\lambda }{ \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} } + \frac{(1-\lambda ) }{ \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}} } \right) = \frac{1}{ {\hat{\mu }}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} } . \end{aligned}$$

Since $\beta _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ is the $\hbox {weak}^*$ limit of $\beta _n$, $\phi _n \rightarrow \phi$ uniformly. Let $\hat{g} = g \circ \Phi ^{-1}$. Note that $\hat{g}_n$ is a bounded sequence in $H^1({\widehat{\Omega }} {\setminus } K_L)$, moreover, $\hat{g}_n \rightarrow \hat{g}$ in $L^2(\partial _D {\widehat{\Omega }})$.

Let $\hat{u}_{l,n} \in \textrm{argmin} \{ \widehat{\mathcal {W}}_{l,n} ( \hat{u} ) : \hat{u} \in {\widehat{\mathcal {U}}}_{l,n} \}$ and $u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \in \textrm{argmin} \{ \mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} ( u ) : u \in \mathcal {U}_{{\hspace{0.5pt}l}} \}$. We know that $\hat{u}_{l,n} \rightharpoonup \hat{u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ in $H^1 ( {\widehat{\Omega }} {\setminus } K_l)$. Since $\hat{\mu }_{n,{\tiny \text {1}}} = \hat{\mu }_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \mu _{{\hspace{-4pt} A}, {\tiny \text {1}}} \, \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ we get

$$\begin{aligned} \int _{{\widehat{\Omega }}_{(a,b)} {\setminus } K_l} \hat{\mu }_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} \, | \partial _{y_1} \hat{u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}y \leqq \liminf _{n \rightarrow \infty } \int _{{\widehat{\Omega }}_{(a,b)} {\setminus } K_l} \hat{\mu }_{n,{\tiny \text {1}}} \, | \partial _{y_1} \hat{u}_{l,n} |^2 \,\textrm{d}y , \end{aligned}$$

where ${\widehat{\Omega }}_{(a,b)} = (a,b) \times (-S, S)$. Clearly, $\hat{u}_{l,n} = u_{l,n} \circ \Phi _n^{-1}$, where $u_{l,n}\! \in \! \textrm{argmin} \{ \mathcal {W}_{l,n} ( u ) : u \!\in \! \mathcal {U}_{{\hspace{0.5pt}l}} \}$ and $\hat{u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \!=\! u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \circ \Phi ^{-1}$ where $u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \in \textrm{argmin} \{ \mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} ( u ) : u \in \mathcal {U}_{{\hspace{0.5pt}l}} \}$. Therefore, applying the changes of variable we get

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} \, | \partial _{x_1} u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x \leqq \liminf _{n \rightarrow \infty } \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,n} |^2 \,\textrm{d}x . \end{aligned}$$

Using a different change of variable we can apply the above argument in such a way that

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} \, | \partial _{x_2} u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x \leqq \liminf _{n \rightarrow \infty } \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {2}}} \, | \partial _{x_2} u_{l,n} |^2 \,\textrm{d}x . \end{aligned}$$

By Lemma 30 we have

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,n} |^2 + \mu _{n,{\tiny \text {2}}} \, | \partial _{x_2} u_{l,n} |^2 \,\textrm{d}x \end{aligned}$$

which converges to

$$\begin{aligned} \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},\,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,\,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 + \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},\,{\tiny \text {2}}} \, | \partial _{x_2} u_{l,\,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x , \end{aligned}$$

and thus

$$\begin{aligned}&\limsup _{n \rightarrow \infty } \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,n} |^2 \,\textrm{d}x\\&\quad = \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},\,{\tiny \text {1}}} \, | \partial _{x_1} u_{l,\,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 + \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} \, | \partial _{x_2} u_{l,\,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x \\&\qquad - \liminf _{n \rightarrow \infty } \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{n,{\tiny \text {2}}} \, | \partial _{x_2} u_{l,n} |^2 \,\textrm{d}x \\&\quad \leqq \int _{\Omega _{(a,b)} {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} \, | \partial _{x_1} u_{l,\,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \,\textrm{d}x . \end{aligned}$$

The proof is concluded. $\square$

Proof of Theorem 27

It is enough to show that $\hspace{0.5pt}\mathcal {G}_n \rightarrow \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ pointwise in (0, L).

Let $r>0$ such that $0< 2r< l < L - 2r$. Let $\phi _r : [0,L] \rightarrow [0,1]$ be the Lipschitz map defined by $\phi _r = 0$ in (0, r) and in $(L- r,L)$, $\phi _r =1$ in $(2r,L-2r)$, $\phi _r$ is affine in (r, 2r) and in $(L-2r, L- r)$. By Lemma 21 we can write

$$\begin{aligned} \mathcal {E}'_n ( l) = \int _{\Omega {\setminus } K_l} \nabla u_{l,n} \varvec{C}_n \varvec{E} \nabla u_{l,n}^{T} \, \phi '_r \textrm{d}x \quad \text {for} \quad \varvec{E} = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right) = \textbf{I} + \left( \begin{matrix} - 2 & 0 \\ 0 & 0 \end{matrix} \right) , \end{aligned}$$

where $\varvec{I}$ denotes the identity matrix; a similar representation holds also for $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$. Then, splitting $\varvec{E}$ as above,

$$\begin{aligned} \hspace{0.5pt}\mathcal {G}_n (l)&= - \mathcal {E}'_n (l) = - \int _{\Omega {\setminus } K_l} \nabla \hat{u}_{l,n} \, \varvec{C}_n \nabla \hat{u}^T_{l,n} \, \phi '_r \, \textrm{d}x + 2 \int _{\Omega {\setminus } K_l} \mu _{n,{\tiny \text {1}}} | \partial _{x_1} {u}_{l,n} |^2 \phi '_r \, \textrm{d}x . \end{aligned}$$

By the definition of $\phi _r$, we can write

$$\begin{aligned} \int _{\Omega {\setminus } K_l} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}^T_{l,n} \, \phi '_r \, \textrm{d}x&= r^{-1} \int _{\Omega _{(r,2r)} {\setminus } K_l } \hspace{-12pt} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}^T_{l,n} \, \textrm{d}x \\&\quad - r^{-1} \int _{\Omega _{(L-2r,L-r)} {\setminus } K_l } \hspace{-12pt} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}^T_{l,n} \, \textrm{d}x . \end{aligned}$$

Hence by Lemma 30 we get

$$\begin{aligned} \int _{\Omega {\setminus } K_l} \nabla {u}_{l,n} \, \varvec{C}_n \nabla {u}^T_{l,n} \, \phi '_r \, \textrm{d}x \ \rightarrow \ \int _{\Omega {\setminus } K_l} \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}^T_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \, \phi '_r \, \textrm{d}x . \end{aligned}$$

In the same way we get

$$\begin{aligned} \int _{\Omega {\setminus } K_l} \mu _{n,{\tiny \text {1}}} | \partial _{x_1} {u}_{l,n} |^2 \phi '_r \, \textrm{d}x \ \rightarrow \ \int _{\Omega {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} | \partial _{x_1} {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} |^2 \phi '_r \, \textrm{d}x . \end{aligned}$$

Hence $\hspace{0.5pt}\mathcal {G}_n (l) \rightarrow \hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$. $\square$

4 Non-monotone Boundary Condition

Consider a Dirichlet boundary condition of the type f(t)g where $f \in C^1( [0,T])$ with $f(0)=0$ (f is not necessarily monotone). We will provide an existence and convergence result.

Let ${\hat{\ell }}_n$ and ${\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ be the unique solutions obtained with the monotone boundary condition $u = s g$ and $s \in [0, S]$, where $S = \max \{ | f ( t) | : t \in [0,T] \}$. By Theorem 12 we know that ${\hat{\ell }}_n \rightarrow {\hat{\ell }}$ pointwise in [0, S) and that ${\hat{\ell }}^+ = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$.

Following [18, Section 12] we introduce the non-decreasing function $\bar{f} (t) = \max \{ | f (\tau ) | : \tau \in [0,t] \}$ and we define $\ell _n = {\hat{\ell }}_n \circ \bar{f}$. We will prove that

• (1) $\ell _n$ is a quasi-static evolution in the sense of Proposition 2, that is, $\ell _n : [0,T) \rightarrow [L_0,L]$ is non-decreasing, right continuous, satisfies the initial condition $\ell _n(0) = L_0$ and Griffith’s criterion, in the following form:

  • (i) $G_n (t, \ell _n(t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n(t))$ for every time $t \in [0,T)$;

  • (ii) if $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t))$ then $\ell _n$ is right differentiable in t and $\dot{\ell }^+_n (t) =0$;

  • (iii) if $t \in J ( \ell _n)$ then $G_n (t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [\ell ^-_n(t) , \ell _n(t) )$;

• 2) $\ell _n = {\hat{\ell }}_n \circ \bar{f}$ converge to $\ell = {\hat{\ell }} \circ \bar{f}$ pointwise in [0, T) (up to subsequences); the evolution $\ell$ is non-decreasing, satisfies the initial condition and Griffith’s criterion, in the following form:

  • (i) $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t, \ell (t) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell (t))$ for every time $t \in [0,T)$;

  • (ii) if $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( t, \ell (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(\ell (t))$ then $\ell$ is right differentiable in t and $\dot{\ell }^+ (t) =0$;

  • (iii) if $t \in J ( \ell )$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(t , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ for every $l \in [ \ell ^- (t) , \ell ^+ (t) )$.

Let us prove (1). Clearly $\ell _n$ is non-decreasing, since both ${\hat{\ell }}_n$ and $\bar{f}$ are non-decreasing, and $\ell _n (0) = {\hat{\ell }}_n \circ f (0) = {\hat{\ell }}_n (0) = L_0$. The right continuity of ${\hat{\ell }}_n$ together with the monotonicity and continuity of $\bar{f}$ imply the right continuity of $\ell _n$. It remains to check Griffith’s criterion, but in this case the representation formula (7) does not hold. Before proceeding, note that for $s = \bar{f} (t)$ we have

$$\begin{aligned} {\hat{\ell }}_n^- (s) \leqq \ell _n^- (t) \leqq \ell _n^+(t) \leqq {\hat{\ell }}_n (s) . \end{aligned}$$

(47)

• (i) We know that $G_n (s , {\hat{\ell }}_n(s) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n ({\hat{\ell }}_n(s))$ for every $s \in [0,S)$; writing $s = \bar{f}(t)$ and $\ell _n = {\hat{\ell }} \circ \bar{f}$ yields

  $$\begin{aligned} \bar{f}^{\,2}(t) \hspace{0.5pt}\mathcal {G}_n ( \ell _n (t) )= & \bar{f}^{\,2}(t) \hspace{0.5pt}\mathcal {G}_n ( {\hat{\ell }}_n \circ \bar{f} (t) )\! =\! G_n ( \bar{f} (t) , {\hat{\ell }}_n \circ \bar{f} (t)) \!\leqq \!G^{\hspace{-2.5pt}{ \textit{c}}}_n ({\hat{\ell }}_n \circ \bar{f} (t) ) \\ & = G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t) ) . \end{aligned}$$

  Since $f^2 \leqq \bar{f}^{\,2}$ we get

  $$\begin{aligned} G_n ( t, \ell _n (t) ) = f^2 (t) \hspace{0.5pt}\mathcal {G}_n ( \ell _n (t) ) \leqq \bar{f}^{\,2} (t) \hspace{0.5pt}\mathcal {G}_n ( \ell _n(t)) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t) ) . \end{aligned}$$

• (ii) If $| f(t) | = \bar{f} (t)$ then $G_n ( t, \ell _n (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\ell _n (t))$ reads as

  $$\begin{aligned} \bar{f}^{\,2} (t) \hspace{0.5pt}\mathcal {G}_n ( {\hat{\ell }}_n \circ \bar{f} (t) ) = f^{2} (t) \hspace{0.5pt}\mathcal {G}_n ( {\hat{\ell }}_n \circ \bar{f} (t) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ({\hat{\ell }}_n \circ \bar{f} (t)) \end{aligned}$$

  which, for $s = \bar{f}(t)$, gives $G_n (s , {\hat{\ell }}_n(s) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n({\hat{\ell }}_n (s))$. Since $\bar{f}$ is continuous and monotone non-decreasing, it turns out that

  $$\begin{aligned} 0 \leqq {\dot{\ell }}^+_n (t) \leqq \lim _{h \rightarrow 0^+} \frac{{\hat{\ell }}_n(s+h) - \hat{\ell }_n(s) }{h} = 0 . \end{aligned}$$

  If $| f(t) | < \bar{f} (t)$ then $\bar{f} (t) = f ( t^*)$ for some $t^* < t$. Moreover, by the continuity of f there exists $\delta >0$ such that $\bar{f} (t') = f ( t^*)$ for every $| t' - t | < \delta$. Hence $\ell _n (t') = {\hat{\ell }}_n \circ \bar{f} (t') = \ell _n \circ f (t^*)$ for every $| t' - t | < \delta$; hence $\ell _n$ is constant in a neighborhood of t.

• (iii) Let $t \in J ( \ell _n)$. First, note that $\bar{f} ( t ) = | f(t )|$; indeed, if $\bar{f} ( t ) > | f(t )|$ then, repeating the argument above, $\ell _n$ would be constant in a neighborhood of t and thus $t \not \in J (\ell _n)$. By (47) it turns out that $s \in J ({\hat{\ell }}_n)$. Hence, $G_n ( s , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every $l \in [ {\hat{\ell }}^-_n (s ) , {\hat{\ell }}^+_n (s ) )$; the substitution $s = \bar{f}(t )$ and (47) lead to

  $$\begin{aligned} G_n (t, l) = f^2 (t) \hspace{0.5pt}\mathcal {G}_n (l) = \bar{f}^{\,2}(t) \hspace{0.5pt}\mathcal {G}_n (l ) = G_n (s, l) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \end{aligned}$$

  for every $l \in [\ell ^-_n (t) , \ell ^+_n(t) )$.

Let us prove 2). Before proceeding, note that in general $\ell ^+ = ({\hat{\ell }} \circ \bar{f})^+ \ne {\hat{\ell }}^+ \circ \bar{f} = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\circ \bar{f}$, therefore we cannot employ directly the properties of ${\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$. Instead, we first check Griffith’s criterion for ${\hat{\ell }}$. We know that ${\hat{\ell }}^+ = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ and thus $J ( {\hat{\ell }} ) = J ( {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}})$, moreover ${\hat{\ell }}^- (s) = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}^- (s)$ and ${\hat{\ell }}^+ (s) = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s)$ for every $s \in J ( {\hat{\ell }} )$, while ${\hat{\ell }} (s) = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s)$ for every $s \in [0,S) {\setminus } J ( {\hat{\ell }})$.

• (iii) By Griffith’s criterion for ${\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$, if $s \in J ( {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}) = J ({\hat{\ell }})$ then $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}(l)$ for every $l \in [ {\hat{\ell }}^-_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s) , {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s) ) = [ {\hat{\ell }}^- (s) , {\hat{\ell }}^+(s) )$.

• (ii) If $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( s , {\hat{\ell }} (s) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( {\hat{\ell }} (s))$ then either $s \not \in J ( {\hat{\ell }})$ or $s \in J({\hat{\ell }})$ and ${\hat{\ell }} (s) = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s)$ (otherwise the opposite inequality would hold). In both the cases ${\hat{\ell }} (s) = {\hat{\ell }}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s)$ and their right derivatives coincide. Therefore, ${\hat{\ell }}$ is right differentiable in s and its right derivative vanishes.

• (i) We employ the definition of $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}$. If $\hspace{0.5pt}\mathcal {G}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( {\hat{\ell }} (s) ) =0$ or $G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( {\hat{\ell }} (s) ) = +\infty$ there is nothing to prove. Otherwise, by pointwise convergence

  $$\begin{aligned} \frac{ G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}( s , {\hat{\ell }} (s)) }{G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( {\hat{\ell }}(s)) } \leqq \liminf _{n \rightarrow \infty } \frac{ G_n ( s , {\hat{\ell }}_n (s)) }{G^{\hspace{-2.5pt}{ \textit{c}}}_n ( {\hat{\ell }}_n (s)) } \leqq 1 , \end{aligned}$$

  which implies that $G_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(s , {\hat{\ell }} (s) ) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_{\hspace{-3.5pt} \textit{ef}\hspace{0.15pt}\textit{f}\hspace{1pt}}( {\hat{\ell }} (s))$ for every $s \in [0,S)$.

At this point, employing the change of variable $s = \bar{f}(t)$ and arguing as in point (1) it follows that Griffith’s criterion holds for $\ell = {\hat{\ell }} \circ \bar{f}$.

Appendices

Appendix A: Balanced Viscosity Solutions and Griffith’s Criterion

For sake of completeness, we provide the notion of balanced viscosity solutions for the rate independent system $(\mathcal {E}_n ,\mathcal {R}_n)$, making reference to [16] with a few differences due to irreversibility and lack of regularity in $\Lambda _n$. Moreover we briefly show that the evolution $\ell _n$ (provided by Proposition 2) is a balanced viscosity solution and, vice versa, that each right continuous balanced viscosity solution satisfies Griffith’s criterion. A similar reasoning applies to $\ell _{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$.

Following [16] a right continuous evolution $\gamma _n$ is a balance viscosity evolution for the rate independnent system $(\mathcal {E}_n , \mathcal {R}_n)$ if it satisfies the following conditions for every time $t \in [0,T]$:

$$\begin{aligned} & \partial _{l'} \mathcal {R}_n ( \gamma _n(t) , 0 ) + \partial _{l} E_n ( t , \gamma _n (t)) \ni 0 \end{aligned}$$

(48)

$$\begin{aligned} & E_n (t, \gamma _n(t)) = E_n ( 0, L_0) + \int _0^t P_n ( t, \gamma _n (t)) \, \textrm{d}t - \textrm{Var}_{\mathfrak {f}_n} ( \gamma _n ; [0,t] ) . \end{aligned}$$

(49)

We recall that the state dependent dissipation $\mathcal {R}_n : [L_0, L] \times \mathbb {R} \rightarrow [0,+\infty ]$ is given by

$$\begin{aligned} \mathcal {R}_n ( l , l' ) = {\left\{ \begin{array}{ll} G^c_n (l) \, l' & l' \geqq 0 , \\ +\infty & l' < 0 . \end{array}\right. } \end{aligned}$$

We denote by $\partial _{l'}\mathcal {R}_n$ the subdifferential of the dissipation $\mathcal {R}_n ( l , \cdot )$ and by $\partial _l E_n ( t , l)$ the partial derivative of the reduced elastic energy. Then $\partial _{l'} \mathcal {R}_n ( l , 0 ) = ( -\infty , G^c_n(l)]$ while $\partial _l E_n ( t , l) = - G_n ( t , l)$, if $l \not \in \Lambda _n$. If $l \in \Lambda _n$ the energy $\mathcal {E}_n$ is not differentiable, therefore following (5) we set $\partial _l E_n ( t , l) = - G_n ( t , l)$.

Next, let us introduce the dissipation “distance” $\textsf{dist}_n : [L_0,L] \times [L_0,L] \rightarrow [0, +\infty ]$ given by

$$\begin{aligned} \textsf{dist}_n ( l_0 , l_{\tiny \text {1}}) = \inf \left\{ \int _0^1 \mathcal {R}_n ( l(r) , l'(r) ) \, \textrm{d}r : l \in AC (0,1) , \, l (0) = l_0 , \, l(1)=l_{\tiny \text {1}}\right\} \end{aligned}$$

and the Finsler dissipation cost $\Delta _{\mathfrak {f}_n } : [0,T] \times [L_0,L] \times [L_0,L] \rightarrow [0,+\infty ]$ given by

$$\begin{aligned} \Delta _{\mathfrak {f}_n } ( t, l_0 , l_{\tiny \text {1}}) = \inf \left\{ \int _0^1 \mathfrak {f}_n ( t , l(r) , l'(r) ) \, \textrm{d}r : l \in AC (0,1) , \, l (0) = l_0 , \, l(1)=l_{\tiny \text {1}}\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathfrak {f}_n ( t , l(r) , l'(r) )&= \mathcal {R}_n ( l(r) , l'(r) ) + \textrm{dist} \big ( \!-\! \partial _l E ( t , l (r) ) , \partial _{l'} \mathcal {R}_n ( l (r) , 0 ) \big ) | {\hspace{0.5pt}l}' (r) | . \end{aligned}$$

Finally, let us introduce the notation

$$\begin{aligned} \textrm{Var}_{\mathfrak {f}_n} ( \gamma _n ; [0, t]) = \textrm{Var}_{\textsf{dist}_n} ( \gamma _n ; [0, t]) - \textrm{Jmp}_{\textsf{dist}_n} ( \gamma _n ; [0, t]) + \textrm{Jmp}_{\mathfrak {f}_n} ( \gamma _n ; [0, t]) , \end{aligned}$$

where

$$\begin{aligned} \textrm{Var}_{\textsf{dist}_n} ( \gamma _n ; [0, t])&= \sup \left\{ \sum _{i=0}^{m-1} \mathsf {dist_n} ( \gamma _n (t_{i+1}) , \gamma _n ( t_i ) ) : 0 = t_0 \leqq \cdots \leqq t_m = t \right\} , \\ \textrm{Jmp}_{\textsf{dist}_n} ( \gamma _n ; [0, t])&= \sum _{\tau \in J ( \gamma _n) \cap [0,t]} \textsf{dist}_n ( \gamma _n (\tau ) , \gamma _n^-(\tau )) , \\ \textrm{Jmp}_{\mathfrak {f}_n} ( \gamma _n ; [0, t])&= \sum _{\tau \in J ( \gamma _n) \cap [0,t]} \Delta _n ( \tau , \gamma _n (t) , \gamma _n^-(t)), \end{aligned}$$

(remember that here $\gamma _n$ is right continuous).

First, let us check that the evolution $\ell _n$ provided by Proposition 2 is a balanced viscosity solution. The inequality $G_n ( t, \ell _n (t) ) \leqq G^c_n ( \ell _n(t))$ reads as

$$\begin{aligned} - \partial E_n ( t , \ell _n (t)) \in (- \infty , G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \ell _n(t)) ] = \partial _{l'} \mathcal {R}_n ( \ell _n(t) , 0 ) , \end{aligned}$$

which is (48). Next, let us check that the energy identity (11) implies (49), showing that

$$\begin{aligned} \textrm{Var}_{\mathfrak {f}_n} ( \ell _n ; [0,t] )&= \textrm{Var}_{\textsf{dist}_n} ( \ell _n ; [0, t]) - \textrm{Jmp}_{\textsf{dist}_n} ( \ell _n ; [0, t]) + \textrm{Jmp}_{\mathfrak {f}_n} ( \ell _n ; [0, t]) , \\&= D_n ( \ell _n(t)) - \hspace{-12pt} \sum _{\tau \, \in \, J(\ell _n) \cap [0,t]} \llbracket F_n ( \tau , \ell _n (\tau )) \rrbracket . \end{aligned}$$

First of all, note that in our irreversible setting it is not restrictive to consider $l' \geqq 0$ in the definition of both $\textsf{dist}_n$ and $\Delta _{\mathfrak {f}_n}$. By explicit computations, it is easy to check that

$$\begin{aligned} \textsf{dist}_n ( l_0 , l_{\tiny \text {1}}) = {\left\{ \begin{array}{ll} D_n ( l_{\tiny \text {1}}) - D_n (l_0) & \text {if } l_0 \leqq l_{\tiny \text {1}}, \\ +\infty & \text {otherwise.} \end{array}\right. } \end{aligned}$$

As a consequence, by monotonicity and right continuity of $\ell _n$,

$$\begin{aligned} \textrm{Var}_{\textsf{dist}_n} ( \ell _n ; [0, t])&= D_n ( \ell _n (t) ) , \\ \textrm{Jmp}_{\textsf{dist}_n} ( \ell _n ; [0, t])&= \sum _{\tau \in J ( \ell _n) \cap [0,t]} \textsf{dist}_n ( \ell _n (t) , \ell _n^-(t)) = \sum _{\tau \in J ( \ell _n) \cap [0,t]} \llbracket D_n (\ell _n (\tau )) \rrbracket . \end{aligned}$$

Next, note that

$$\begin{aligned} \textrm{dist} \big ( \!-\! \partial _l E ( t , l) , \partial _{l'} \mathcal {R}_n ( l , 0 ) \big ) = [ G (t, l) - G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) ]_+ \end{aligned}$$

By condition (iii) of Griffith’s criterion for $\tau \in J ( \ell _n)$ we have $G_n (\tau , l ) - G^{\hspace{-2.5pt}{ \textit{c}}}_n (l) \geqq 0$ for every $l \in [ \ell _n ^-(\tau ) , \ell _n (\tau ))$, and thus

$$\begin{aligned} \mathfrak {f}_n ( \tau , l(r) , l'(r) ) = G_n (\tau , l (r) ) \, {\hspace{0.5pt}l}' (r) \end{aligned}$$

for every monotone $l \in AC(0,1)$ with $l(0) = \ell _n^- (\tau )$ and $l(1) = \ell _n (\tau )$. It follows easily that

$$\begin{aligned} \Delta _n ( \tau , \ell _n (\tau ) , \ell _n^-(\tau )) = - \llbracket E_n (\tau , \ell _n (\tau )) \rrbracket , \end{aligned}$$

and thus

$$\begin{aligned} \textrm{Jmp}_{\mathfrak {f}_n} ( \ell _n ; [0, t]) = \sum _{\tau \in J ( \ell _n) \cap [0,t]} - \llbracket E_n (\tau , \ell _n (\tau )) \rrbracket . \end{aligned}$$

Conversely, let $\gamma _n$ be a (right continuous) balanced viscosity solution. If $\gamma _n$ satisfies (48) then

$$\begin{aligned} (- \infty , G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \gamma _n(t) ) ] - G_n ( t , \gamma _n (t)) \ni 0 , \end{aligned}$$

that is $G_n ( t , \gamma _n (t)) \leqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (\gamma _n(t))$. Let us check that $G_n ( \tau , l ) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for $\tau \in J (\gamma _n)$ and every $l \in [ \gamma _n^- (\tau ) , \gamma _n (\tau )]$. In the interval $[ \tau -\delta , \tau + \delta ]$ (for some $\delta >0$) the energy identity (49) reads as

$$\begin{aligned} & E_n (\tau + \delta , \gamma _n(\tau +\delta )) - E_n ( \tau -\delta , \gamma _n (\tau - \delta ) ) = \int _{\tau - \delta }^{\tau + \delta } P_n ( t, \gamma _n (t)) \, \textrm{d}t \\ & \qquad - \textrm{Var}_{\mathfrak {f}_n} ( \gamma _n ; [\tau -\delta ,\tau +\delta ] ) . \end{aligned}$$

Hence, taking the limit as $\delta \rightarrow 0^+$, we get

$$\begin{aligned} E_n ( \tau , \gamma _n^- (\tau ) ) - E_n ( \tau , \gamma _n (\tau )) = \int _{\gamma _n^- (\tau )}^{\gamma _n (\tau )} G_n ( \tau , s) \, \textrm{d}s = \lim _{\delta \rightarrow 0^+} \textrm{Var}_{\mathfrak {f}_n} ( \gamma _n ; [\tau -\delta ,\tau +\delta ] ) . \end{aligned}$$

(50)

It is easy to check that

$$\begin{aligned} \lim _{\delta \rightarrow 0^+} \textrm{Var}_{\mathfrak {f}_n} ( \gamma _n ; [\tau -\delta ,\tau +\delta ] )&= \Delta ( \tau , \gamma _n^- (\tau ) , \gamma _n (\tau )) \\&= \int _{\gamma _n^- (\tau )}^{\gamma _n (\tau )} G^{\hspace{-2.5pt}{ \textit{c}}}_n ( s) + [ G_n ( \tau , s) - G^{\hspace{-2.5pt}{ \textit{c}}}(s) ]_+ \textrm{d}s . \end{aligned}$$

Hence, by (50),

$$\begin{aligned} \int _{\gamma _n^- (\tau )}^{\gamma _n (\tau )} G_n ( \tau , s) - G^{\hspace{-2.5pt}{ \textit{c}}}_n ( s) \, \textrm{d}s = \int _{\gamma _n^- (\tau )}^{\gamma _n (\tau )} [ G_n ( \tau , s) - G^{\hspace{-2.5pt}{ \textit{c}}}(s) ]_+ \textrm{d}s , \end{aligned}$$

which in turn implies $G_n ( \tau , s) - G^{\hspace{-2.5pt}{ \textit{c}}}_n (s) \geqq 0$ for almost everywhere $s \in (\gamma _n^- (\tau ) , \gamma _n (\tau ))$. By the regularity of energy release and toughness we conclude that $G_n ( \tau , s) \geqq G^{\hspace{-2.5pt}{ \textit{c}}}_n (s)$ holds for every $s \in [\gamma _n^- (\tau ) , \gamma _n (\tau ))$. It remains to show that $G_n (\tau , \gamma _n(\tau )) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \gamma _n(\tau ))$ implies $\dot{\gamma }_n^+ (\tau ) =0$. Let us first assume that $\ell _n (\tau ) \not \in \Lambda _n$. Then, by continuity of $G_n$ and $G^{\hspace{-2.5pt}{ \textit{c}}}_n$ it follows that $G_n ( t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ for every (t, l) such that $| \tau -t | < \delta$ and $| l - \ell _n(\tau ) | < \delta$. By right continuity we can further assume that $| \gamma _n (t) - \gamma _n (\tau ) | < \delta$ for $t \in ( \tau , \tau +\delta )$. The fact that $G_n ( t , l) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (l)$ implies that there are no jumps in $t \in ( \tau , \tau +\delta )$, otherwise (iii) (which we already proved) would be violated. Hence for $t \in ( \tau , \tau +\delta )$ the energy identity reads as

$$\begin{aligned} E_n (t, \gamma _n(t) ) = E_n ( \tau , \gamma _n (\tau )) + \int _{\tau }^{t} P_n (t, \gamma _n (t)) \, \textrm{d}t - \textrm{Var}_{\textsf{dist}_n} ( \gamma _n ; [\tau , t]) \end{aligned}$$

Writing

$$\begin{aligned} E_n (t, \gamma _n(t) ) - E_n ( \tau , \gamma _n (\tau ))&= \int _{\tau }^{t} \partial _t E_n (t, \gamma _n (t)) \, \textrm{d}t + \int _\tau ^t \partial _l E_n (t, \gamma _n (t)) \, d\gamma _n (t) \\ \int _{\tau }^{t} P_n (t, \gamma _n (t)) \, \textrm{d}t&= \int _{\tau }^{t} \partial _t E_n (t, \gamma _n (t)) \, \textrm{d}t \\ \textrm{Var}_{\textsf{dist}_n} ( \gamma _n ; [\tau , t])&= \int _{\tau }^t G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \gamma _n (t) ) \, d\gamma _n (t), \end{aligned}$$

it follows that

$$\begin{aligned} \int _\tau ^t G_n (t, \gamma _n (t)) \, d\gamma _n (t) = \int _{\tau }^t G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \gamma _n (t) ) \, d\gamma _n (t) . \end{aligned}$$

This is a contradiction with $G_n (t, \gamma _n (t)) < G^{\hspace{-2.5pt}{ \textit{c}}}_n ( \gamma _n (t) )$ unless $d\gamma _n = 0$, which implies the thesis. It $\gamma _n (\tau ) \in \Lambda _n$ then there exists a sequence $l_m \searrow \gamma _n (\tau )$ such that $l_m \not \in \Lambda _n$ and $G_n (\tau , l_m ) \rightarrow G_n ( \tau , \gamma _n (\tau ) ) < G^{\hspace{-2.5pt}{ \textit{c}}}_n (\gamma _n (\tau ) ) = G^{\hspace{-2.5pt}{ \textit{c}}}_n ({\hspace{0.5pt}l}_m)$. Since $l_m \not \in \Lambda _n$, arguing as above it follows that $\gamma _n (t) < l_m$ in a right neighborhood of $\tau$, uniformly with respect to $m \in \mathbb {N}$. Hence, taking the infimum with respect to $m \in \mathbb {N}$, we get $\gamma _n (t) \leqq \gamma _n (\tau )$; by monotonicity, $\gamma _n$ is constant in a right neighborhood of $\tau$.

Homogenization of Displacement, Stress, and Energy

This section contains classic results on the homogenization of Dirichlet energies (see for example [17] or [9] and the references therein). We provide a short proof, adapted to the case of horizontal layers. We recall that

$$\begin{aligned} \, \varvec{C}_{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} = \left( \begin{matrix} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {1}}} & 0 \\ 0 & \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} \end{matrix} \right) , \end{aligned}$$

where $1 / \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}} = \lambda / \mu _{{\hspace{-4pt} A},{\tiny \text {1}}} + (1-\lambda ) / \mu _{{\hspace{-3.5pt} B},{\tiny \text {1}}}$ and $\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} = \lambda \mu _{{\hspace{-4pt} A},{\tiny \text {2}}} + ( 1- \lambda ) \mu _{{\hspace{-3.5pt} B},{\tiny \text {2}}}$. For $l \in [0,L]$ we denote

$$\begin{aligned} u_{l,n} \in \textrm{argmin} \,\{ \mathcal {W}_{l,n} ( u ) : u \in \mathcal {U}_l \} , \quad u_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \in \textrm{argmin} \,\{ \mathcal {W}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} ( u ) : u \in \mathcal {U}_l \} . \end{aligned}$$

Proposition 32

Let $l \in [0,L]$. Then $u_{l,n} \rightharpoonup {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ in $H^1( {\Omega } {\setminus } K_l)$ and $\, \varvec{C}_n \nabla {u}_{l,n}^T \rightharpoonup \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T$ in $L^2 ( {\Omega } {\setminus } K_l ; \mathbb {R}^2)$.

Proof

By uniform coercivity ${u}_{l,n}$ is a bounded sequence in $H^1( {\Omega } {\setminus } K_l)$ and thus $\, \varvec{C}_n \nabla {u}_{l,n}^T$ is a bounded sequence in $L^2 ( {\Omega } {\setminus } K_l ; \mathbb {R}^2)$. Then, up to non-relabeled subsequences, ${u}_{l,n} \rightharpoonup {u}_{l,\infty }$ in $H^1 ( {\Omega } {\setminus } K_l)$ while $\, \varvec{C}_n \nabla {u}_{l,n}^T \rightharpoonup \varvec{\sigma }_{l,\infty }$ in $L^2 ( {\Omega } {\setminus } K_l ; \mathbb {R}^2)$. We will prove that

$$\begin{aligned} \varvec{\sigma }_{l,\infty } = \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,\infty }^T . \end{aligned}$$

(51)

From this identity it follows that ${u}_{l,\infty } = {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$; indeed, $u_{l,\infty } \in \mathcal {U}_l$ and, passing to the limit in the variational formulation

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} \nabla v \, \varvec{C}_n \nabla {u}_{l,n} ^T\, \textrm{d}x = 0 \quad \text {for every }v \in \mathcal {V}_l, \end{aligned}$$

we easily get that ${u}_{l,\infty }$ is the unique solution of the variational problem

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} \nabla v \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,\infty } ^T\, \textrm{d}x = 0 \quad \text {for every }v \in \mathcal {V}_l. \end{aligned}$$

The idea to prove (51) is to show that $(\varvec{\sigma }_{l,\infty })_i = \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, i} \partial _{x_i} {u}_{l,\infty }$ for $i=1,2$, by passing to the limit in the constitutive relation, that is, $(\varvec{\sigma }_{l,n} )_{i} = \mu _{n,i} \partial _{x_{i}} {u}_{l,n}$ for $i=1,2$.

I. Let $i=1$. For every $\varphi \in C^\infty _c ( {\Omega } {\setminus } K_l)$ we have

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} ( \varvec{\sigma }_{l,n})_{{\tiny \text {1}}} \varphi / \mu _{n,{\tiny \text {1}}} \, \textrm{d}x = \int _{ {\Omega } {\setminus } K_l} ( \partial _{x_{1}} {u}_{l,n} ) \varphi \, \textrm{d}x . \end{aligned}$$

By weak convergence we can pass to the limit in the right hand side, getting

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} ( \partial _{x_{1}} {u}_{l,n} ) \varphi \, \textrm{d}x \rightarrow \int _{ {\Omega } {\setminus } K_l} ( \partial _{x_{1}} {u}_{l,\infty } ) \varphi \, \textrm{d}x . \end{aligned}$$

Let us prove that

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} ( \varvec{\sigma }_{l,n})_{{\tiny \text {1}}} \varphi / \mu _{n,{\tiny \text {1}}} \, \textrm{d}x \rightarrow \int _{ {\Omega } \setminus K_l} ( \varvec{\sigma }_{l,\infty })_{{\tiny \text {1}}} \varphi / \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {1}}} \, \textrm{d}x . \end{aligned}$$

(52)

Let $I= (-H,0) \cup (0,H)$ and write

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} ( \varvec{\sigma }_{l,n})_{{\tiny \text {1}}} \varphi / \mu _{n,{\tiny \text {1}}} \, \textrm{d}x&= \int _{(0,L)} \Big ( \int _I ( \varvec{\sigma }_{l,n} )_{{\tiny \text {1}}} (x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}} ) \varphi ( x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}}) \, \textrm{d} x_{{\tiny \text {2}}} \Big ) \Big / \mu _{n,{\tiny \text {1}}} ( x_{{\tiny \text {1}}}) \, \textrm{d} x_{{\tiny \text {1}}} \nonumber \\&= \int _{(0,L)} s_n (x_{{\tiny \text {1}}}) / \mu _{n,{\tiny \text {1}}} ( x_{{\tiny \text {1}}} ) \, \textrm{d} x_{{\tiny \text {1}}} , \end{aligned}$$

(53)

where we have introduced the auxiliary function $s_n$ given by

$$\begin{aligned} s_n (x_{1}) = \int _I ( \varvec{\sigma }_{l,n} )_{{\tiny \text {1}}} (x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}} ) \varphi ( x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}}) \, \textrm{d} x_{{\tiny \text {2}}} , \quad \text {for almost\,everywhere }x_{\tiny \text {1}}\in (0,L). \end{aligned}$$

We claim that $s_n \rightarrow s_\infty$ strongly in $L^1 (0,L)$, where

$$\begin{aligned} s_\infty (x_{{\tiny \text {1}}}) = \int _I (\varvec{\sigma }_{l,\infty })_{{\tiny \text {1}}} ( x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}} ) \, \varphi ( x_{{\tiny \text {1}}} , x_{{\tiny \text {2}}} ) \, \textrm{d} x_{{\tiny \text {2}}} \end{aligned}$$

from which (52) follows from (53) because $1/ \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {1}}}$ is the $\hbox {weak}^*$ limit of $1/ \mu _{n,{\tiny \text {1}}}$ in $L^\infty (0,L)$.

Denote by $\sigma _n$ the (vector valued) function $\sigma _{n} (x_{{\tiny \text {1}}} ) = (\varvec{\sigma }_{l,n})_{{\tiny \text {1}}} ( x_{{\tiny \text {1}}} , \cdot )$; since $\varvec{\sigma }_n$ is bounded in $L^2( {\Omega } {\setminus } K_l)$, it turns out that $\sigma _n$ is bounded in $L^2( 0, L; L^2(I) )$. We claim that $\sigma _n'$ is bounded in $L^2( 0, L ; H^{-1} (I) )$. Let us start writing

$$\begin{aligned} \Vert \sigma _n' \Vert _{L^2(0,L;H^{-1}(I) )} = \sup \int _{(0,L)} \langle \sigma '_n , z \rangle \, \textrm{d} x_{{\tiny \text {1}}} , \end{aligned}$$

where the supremum is taken over all the (vector valued) functions z with $\Vert z \Vert _{L^2(0,L;H^1_0(I) )} \leqq 1$ and $\langle \cdot , \cdot \rangle$ denotes the duality between $H^{-1} (I)$ and $H^1_0 (I)$. The set of functions of the form $z = \phi \psi$ for $\phi \in C^\infty _c (0,L)$ and $\psi \in C^\infty _c ( I )$ is dense in $L^2(0,L; H^1_0 (I) )$. For any such z we can write

$$\begin{aligned} \int _{(0,L)} \langle \sigma '_n , z \rangle \, \textrm{d} x_{{\tiny \text {1}}}&= \int _{(0,L)} \langle \sigma '_n , \psi \rangle \phi \, \textrm{d}x_{\tiny \text {1}}= - \int _{(0,L)} \langle \sigma _n , \psi \rangle \phi ' \, \textrm{d}x_{\tiny \text {1}}\\&= - \int _{(0,L)} \int _{I} \sigma _n (x_{{\tiny \text {1}}}, x_{{\tiny \text {2}}}) \psi (x_{{\tiny \text {2}}}) \partial _{x_1} \phi (x_{{\tiny \text {1}}}) \, \textrm{d}x_{\tiny \text {2}}\, \textrm{d}x_{\tiny \text {1}}\\&= -\int _{ {\Omega } {\setminus } K_l} \varvec{\sigma }_{l,n} \partial _{x_1} z \, \textrm{d}x \leqq \Vert \varvec{\sigma }_{l,n} \Vert _{L^2 ( {\Omega } {\setminus } K_l)} . \end{aligned}$$

Then, by Schauder’s theorem and Aubin-Lions lemma there exists a subsequence (not relabeled) such that $\sigma _n \rightarrow \sigma _\infty$ (strongly) in $L^2(0,L; H^{-1} (I) )$. As the stresses are $L^2$ functions, writing

$$\begin{aligned} s_n (x_{{\tiny \text {1}}}) = \!\! _{H^{-1}(I) } \langle \sigma _n ( x_{{\tiny \text {1}}} ) , \varphi ( x_{\tiny \text {1}}, \cdot ) \rangle _{H^1_0(I) } \quad \text {and} \quad s_\infty (x_{{\tiny \text {1}}}) = \!\!_{H^{-1}(I) } \langle \sigma _\infty ( x_{{\tiny \text {1}}} ) , \varphi ( x_{\tiny \text {1}}, \cdot ) \rangle _{H^1_0(I) } \end{aligned}$$

it follows that $s_n \rightarrow s$ in $L^1(0,L)$.

II. Let $i=2$. Since $\mu _{n,{\tiny \text {2}}}$ is independent of $x_{\tiny \text {2}}$, for every $\varphi \in C^\infty _c ( {\Omega } {\setminus } K_l)$ we have

$$\begin{aligned} \int _{ {\Omega } {\setminus } K_l} ( \varvec{\sigma }_{l,n})_{{\tiny \text {2}}} \, \varphi \, \textrm{d}x = \int _{ {\Omega } {\setminus } K_l} ( \partial _{x_{2}} ( \mu _{n,{\tiny \text {2}}} {u}_{l,n} ) ) \varphi \, \textrm{d}x = - \int _{ {\Omega } {\setminus } K_l} \mu _{n, {\tiny \text {2}}} {u}_{l,n} \partial _{x_{2}} \varphi \, \textrm{d}x . \end{aligned}$$

We have $\mu _{n,{\tiny \text {2}}} \, \partial _{x_{2}} \varphi {\mathop {\rightharpoonup }\limits ^{*}}\mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}, {\tiny \text {2}}} \, \partial _{x_{2}} \varphi$ in $L^\infty ( {\Omega } {\setminus } K_l)$ and $u_{l,n} \rightarrow {u}_{l,\infty }$ in $L^1 ( {\Omega } {\setminus } K_l)$ (by compact embedding), therefore

$$\begin{aligned}&- \int _{ {\Omega } {\setminus } K_l} \mu _{n,{\tiny \text {2}}} {u}_{l,n} \partial _{x_{2}} \varphi \, \textrm{d}x \rightarrow - \int _{ {\Omega } {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} \partial _{x_{2}} \varphi \, \textrm{d}x\\ &\quad =\int _{ {\Omega } {\setminus } K_l} \mu _{{\hspace{-3pt} \textit{hom}\hspace{0.2pt}},{\tiny \text {2}}} (\partial _{x_{2}} {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}} ) \varphi \, \textrm{d}x \\&\quad = \int _{ {\Omega } {\setminus } K_l} ( \varvec{\sigma }_{{\hspace{0.5pt}l},{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}})_{{\tiny \text {2}}} \varphi \, \textrm{d}x . \end{aligned}$$

The proof is concluded. $\square$

By Proposition 32 and standard arguments it follows the convergence of the reduced energies, that is, $\mathcal {E}_n \rightarrow \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ pointwise in [0, L]. Further, it is not difficult to also prove the following result:

Corollary 33

Let $l_n \in [0,L]$ such that $l_n \rightarrow l$. Then $\nabla u_{l_n,n} \rightharpoonup \nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}$ and $\, \varvec{C}_n \nabla {u}_{l_n,n}^T \rightharpoonup \, \varvec{C}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}\nabla {u}_{l,{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}}^T$ in $L^2 ( {\Omega } ; \mathbb {R}^2)$. Moreover $\mathcal {E}_n ( l_n) \rightarrow \mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}(l)$.

As by-product of the previous Corollary the energy $\mathcal {E}_n$ converge uniformly to $\mathcal {E}_{\hspace{-3pt} \textit{hom}\hspace{0.2pt}}$ in [0, L].