DISCONTINUOUS FINITE ELEMENT APPROXIMATION OF QUASISTATIC CRACK GROWTH IN FINITE ELASTICITY

ALESSANDRO GIACOMINI AND MARCELLO PONSIGLIONE

Abstract. We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed in [13] by G. Dal Maso, G.A. Francfort, and R. Toader, which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we prove also a convergence result for the deformations.

Keywords : variational models, energy minimization, free discontinuity problems, crack propagation, quasistatic evolution, brittle fracture, finite elements. 2000 Mathematics Subject Classification: 35R35, 35J25, 74R10, 35A35, 65L60.

Contents

1. Introduction 1 2. Notations and Preliminaries 4 3. The quasistatic crack growth of Dal Maso-Francfort-Toader 6 4. The finite element space and the transfer of jump 12 5. Preexisting cracks and their approximation 15 6. The discontinuous finite element approximation 17 7. The approximation result 21 8. The strictly convex case 29 References 31

1. Introduction

The aim of this paper is to provide a discontinuous finite element approximation of a model of quasistatic growth of brittle fractures in finite elasticity recently proposed in [13] by Dal Maso, Francfort, and Toader in the framework of the variational theory of crack propagation proposed by Francfort and Marigo in [15]. This theory is inspired to Griffith’s criterion and determines the crack path through a competition between bulk and surface energies.

In the case of linearized elasticity, a first precise mathematical formulation of the model [15] has been given by Dal Maso and Toader [12]: they treat the case of anti-planar shear in dimension two assuming that the fractures are compact sets with a finite number of connected components. This analysis has been extended to the case of plane elasticity by Chambolle in [9]. Francfort and Larsen [14], using the framework of SBV functions (see Section 2), proved the existence of a quasistatic growth of brittle fractures in the case of anti-planar shear in any dimension N ≥ 2 and without assumptions on the structure of the fractures which are dealt with the set of jumps of the displacements. Approximation results for the quasistatic evolution of [14] has been given in [16] and in [17] and provide a theoretical basis to the numerical study of the model given in [5].

The quasistatic crack growth proposed by Dal Maso, Francfort, and Toader in [13] consider the case of finite elasticity, and takes into account possible volume and traction forces applied to the elastic body. In order to describe the result of [13] (a complete description is given in Section 3), let us assume that the elastic body has a reference configuration given by Ω ⊆ ℝ^{N} open, bounded and with Lipschitz boundary. Let ∂_{D}Ω ⊆ ∂Ω be open in the relative topology, and let ∂_{N}Ω := ∂Ω \ ∂_{D}Ω. Let Ω_{B} ⊆ Ω, and let ∂_{S}Ω ⊆ ∂_{N}Ω be such that \overline{Ω}_{B} ∩ ∂_{S}Ω = ∅. Ω_{B} is the brittle part of Ω, and ∂_{S}Ω is the part of the boundary where traction forces are supposed to act. A crack is given by any rectifiable set in \overline{Ω}_{B} with finite (N − 1) Hausdorff measure. Given a boundary deformation g on ∂_{D}Ω and a crack Γ, the family of all admissible deformation of Ω is given by the set AD(g, Γ) of all function u ∈ GSBV (Ω; ℝ^{N}) (see Section 2) such that S(u) ⊆ Γ and u = g on ∂_{D}Ω \ Γ. Here S(u) denotes the set of jumps of u, and the equality u = g is intended in the sense of traces. Requiring u = g only on ∂_{D}Ω \ Γ means that the deformation is assumed not to be transmitted through the fracture. The bulk energy considered in [13] is of the form

∫_{Ω} W (x, ∇u(x)) dx,

where W (x, ξ) is quasiconvex in ξ, and satisfies suitable regularity and growth assumptions (see (3.4) and (3.5)). Moreover the time dependent body and traction forces are supposed to be conservative with work given by

− ∫_{Ω\Γ} F (t, x, u(x)) dx − ∫_{∂_{S}Ω} G(t, x, u(x)) dH^{N−1}(x),

where F and G satisfy suitable regularity and growth conditions (see Section 3). Finally the work made to produce the crack Γ is given by

E^{s}(Γ) := ∫_{Γ} k(x, ν(x)) dH^{N−1}(x),

where ν(x) is the normal to Γ at x, and k(x, ν) satisfies standard hypotheses which guarantee lower semicontinuity (see Section 3). Clearly, W, F, G and k depend on the material. Let us set

E^{el}(t)(u) := ∫_{Ω} W (x, ∇u(x)) dx − ∫_{Ω\Γ} F (t, x, u(x)) dx − ∫_{∂_{S}Ω} G(t, x, u(x)) dH^{N−1}(x),

and

(1.1)  E(t)(u, Γ) := E^{el}(t)(u) + E^{s}(Γ).

Given a boundary deformation g(t) with t ∈ [0, T] and a preexisting crack Γ_{0}, a quasistatic crack growth relative to g and Γ_{0} is a map {t → (u(t), Γ(t)) : t ∈ [0, T]} such that the following conditions hold:

(1) for all t ∈ [0, T]: u(t) ∈ AD(g(t), Γ(t));

(2) irreversibility: Γ_{0} ⊆ Γ(s) ⊆ Γ(t) for all 0 ≤ s ≤ t ≤ T;

(3) static equilibrium: for all t ∈ [0, T] and for all admissible configurations (u, Γ) with Γ(t) ⊆ Γ

E(t)(u(t), Γ(t)) ≤ E(t)(u, Γ);

(4) nondissipativity: the time derivative of the total energy E(t)(u(t), Γ(t)) is equal to the power of external forces (see (3.21)).

In this paper we discretize the model using a suitable finite element method and prove its convergence to this notion of quasistatic crack growth. We restrict our analysis to a two dimensional setting considering only a polygonal reference configuration Ω ⊆ ℝ^{2}.

The discretization of the domain Ω is carried out as in [17] employing adaptive triangulations introduced by M. Negri in [19] (see also [20]). Let us fix two parameters ε > 0 and a ∈]0, \frac{1}{2}[. We consider a regular triangulation R_{ε} of size ε of Ω, i.e. we assume that there exist two constants c_{1} and c_{2} so that every triangle T ∈ R_{ε} contains a ball of diameter c_{1}ε and is contained in a ball of diameter c_{2}ε. In order to treat the boundary data, we assume also that ∂_{D}Ω is composed of edges of R_{ε}. On each edge [x, y] of R_{ε} we consider a point z such that z = tx+(1−t)y with t ∈ [a, 1−a]. These points are called adaptive vertices. Connecting together the adaptive vertices, we divide every T ∈ R_{ε} into four triangles. We take the new triangulation T obtained after this division as the discretization of Ω. The family of all such triangulations will be denoted by T_{ε,a}(Ω).

The discretization of the energy functional is obtained restricting the total energy (1.1) to the family of functions u which are affine on the triangles of some triangulation T(u) ∈ T_{ε,a}(Ω) and are allowed to jump across the edges of T(u) contained in \overline{Ω}_{B}. We indicate this space by A_{ε,a}^{B}(Ω; ℝ^{2}). The boundary data is assumed to belong to the space AF_{ε}(Ω; ℝ^{2}) of continuous functions which are affine on every triangle T ∈ R_{ε}.

Given the boundary data g_{ε} ∈ W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})) with g_{ε}(t) ∈ AF_{ε}(Ω; ℝ^{2}) for all t ∈ [0, T] (p, q are related to the growth assumptions on W, F, G) and an initial crack Γ_{ε,a}^{0} (see Section 6), we divide [0, 1] into subintervals [t_{i}^{δ}, t_{i+1}^{δ}] of size δ > 0 for i = 0, …, N_{δ}, and for all u ∈ A_{ε,a}^{B}(Ω; ℝ^{2}) we indicate by S_{D}^{g_{ε}(t)}(u) the edges of the triangulation T(u) contained in ∂_{D}Ω on which u ≠ g_{ε}(t). Using a variational argument (Proposition 6.1), we construct a discrete evolution {(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) : i = 0, …, N_{δ}} such that for all i = 0, …, N_{δ} we have u_{ε,a}^{δ,i} ∈ A_{ε,a}^{B}(Ω; ℝ^{2}),

Γ_{ε,a}^{δ,i} := ⋃_{r=0}^{i} [S(u_{ε,a}^{δ,r}) ∪ S_{D}^{g_{ε}(t_{r}^{δ})}(u_{ε,a}^{δ,r})],

and the following unilateral minimality property holds: for all v ∈ A_{ε,a}^{B}(Ω; ℝ^{2})

(1.2)  E^{el}(t_{i}^{δ})(u_{ε,a}^{δ,i}) ≤ E^{el}(t_{i}^{δ})(v) + E^{s} ((S(v) ∪ S_{D}^{g_{ε}(t_{i}^{δ})}(v)) \ Γ_{ε,a}^{δ,i−1}) .

Notice that by construction u_{ε,a}^{δ,i} ∈ AD(g_{ε}(t_{i}^{δ}), Γ_{ε,a}^{δ,i}). Moreover the definition of the discrete fracture ensures that Γ_{ε,a}^{δ,i} ⊆ Γ_{ε,a}^{δ,j} for all i ≤ j, recovering in this discrete setting the irreversibilty of the crack growth given in (2). The minimality property (1.2) is the reformulation in the finite element space of the equilibrium condition (3). Finally we obtain an estimate from above for E(t_{i}^{δ})(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) (see Proposition 6.2) which is a discrete version of (4).

In order to perform the asymptotic analysis of the discrete evolution {(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) : i = 0, …, N_{δ}} we make the piecewise constant interpolation in time u_{ε,a}^{δ}(t) = u_{ε,a}^{δ,i} and Γ_{ε,a}^{δ}(t) = Γ_{ε,a}^{δ,i} for all t_{i}^{δ} ≤ t < t_{i+1}^{δ}. Let us suppose that

g_{ε} → g strongly in W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}))

(where on W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}) we take the norm ∥u∥ := ∥u∥_{W^{1,p}(Ω;ℝ^{2})} + ∥u∥_{L^{q}(Ω;ℝ^{2})}), and that Γ_{ε,a}^{0} approximate an initial crack Γ^{0} in the sense of Proposition 5.1.

The main result of the paper (Theorem 7.1) states that there exist a quasistatic evolution {t → (u(t), Γ(t)) : t ∈ [0, T]} in the sense of [13] relative to the boundary deformation g and the preexisting crack Γ^{0} and sequences δ_{n} → 0, ε_{n} → 0, a_{n} → 0, such that setting

u_{n}(t) := u_{ε_{n},a_{n}}^{δ_{n}}(t), Γ_{n}(t) := Γ_{ε_{n},a_{n}}^{δ_{n}}(t),

for all t ∈ [0, T] the following facts hold:

(a) (u_{n}(t))_{n∈ℕ} is weakly precompact in GSBV_{q}^{p}(Ω; ℝ^{2}), and every accumulation point ũ(t) is such that ũ(t) ∈ AD(g(t), Γ(t)), and (ũ(t), Γ(t)) satisfy the static equilibrium (2); moreover there exists a subsequence (δ_{n_{k}}, ε_{n_{k}}, a_{n_{k}})_{k∈ℕ} of (δ_{n}, ε_{n}, a_{n})_{n∈ℕ} (depending on t) such that

u_{n_{k}}(t) ⇀ u(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2})

(see Section 2 for a precise definition of GSBV_{q}^{p}(Ω; ℝ^{2}), and of weak convergence in this space);

(b) convergence of the total energy holds, and more precisely elastic and surface energies converge separately, that is

E^{el}(t)(u_{n}(t)) → E^{el}(t)(u(t)), E^{s}(Γ_{n}(t)) → E^{s}(Γ(t)).

By point (a), the approximation of the deformation u(t) is available only up to a subsequence depending on t: this is due to the possible non uniqueness of the minimum energy deformation associated to Γ(t). In the case E^{el}(t)(u) is strictly convex, it turns out that the deformation u(t) is uniquely determined, and we prove that (Theorem 8.1)

∇u_{n}(t) → ∇u(t) strongly in L^{p}(Ω; M^{2×2}),

and

u_{n}(t) → u(t) strongly in L^{q}(Ω; ℝ^{2}).

The main difficulty to prove Theorem 7.1 consists in passing to the limit in the static equilibrium (1.2). In order to find the fracture Γ(t) in the limit, in Lemma 7.2 and Lemma 7.4 we adapt to the context of finite elements the notion of σ^{p}- convergence of sets formulated in [13]. This is the key tool to obtain the convergence of elastic and surface energies at all times t ∈ [0, T] (while in [17] this was available only at the continuity points of H^{1}(Γ(t))). In order to infer the static equilibrium of Γ(t) from that of Γ_{n}(t), we employ a generalization of the piecewise affine transfer of jumps [17, Proposition 5.1] (see Proposition 4.2).

The paper is organized as follows. In Section 2 we introduce the basic notation, and some tools employed throughout the paper. In Section 3 we describe the quasistatic crack growth of [13] precising the functional setting and the hypotheses on the elastic and surface energies involved. In Section 4 we introduce the finite element space, and in Section 5 we prove an approximation result for a preexisting crack configuration. In Section 6 we prove the existence of a discrete evolution, and in Section 7 we prove the main approximation result (Theorem 7.1). In Section 8 we treat the case of strictly convex total energy.

2. Notations and Preliminaries

In this section we introduce the main notations and the preliminary results employed in the rest of the paper.

Basic notation. We will employ the following basic notation:

- M^{n×m} is the space of n × m matrices;

- H^{1} is the one-dimensional Hausdorff measure;

- for p ∈ [1, +∞], ∥ · ∥_{p} denotes the usual L^{p} norm;

- if μ is a measure on ℝ^{2} and A is a Borel subset of ℝ^{2}, μ A denotes the restriction of μ to A, i.e. (μ A)(B) := μ(B ∩ A) for all Borel sets B ⊆ ℝ^{2};

- if A, B ⊆ ℝ^{2}, A ˜⊂ B means that A ⊆ B up to a set of H^{1}-measure zero.

SBV and GSBV spaces. Let A be an open subset of ℝ^{n}, and let u : A → ℝ^{m} be a measurable function. Given x ∈ A, we say that ũ(x) is the approximate limit of u at x, and we write ũ(x) = ap lim_{y→x} u(y), if for every ε > 0

lim_{r→0} r^{−n}L^{n} ({y ∈ B_{r}(x) : |u(y) − ũ(x)| > ε}) = 0.

Here B_{r}(x) denotes the ball of center x and radius r. We indicate by S(u) the set of points where the approximate limit of u does not exist. We say that the matrix m×n ∇u(x) is the approximate gradient of u at x if

ap lim_{y→x} \frac{u(y) − u(x) − ∇u(x)(y − x)}{|y − x|} = 0.

We say that u ∈ BV (A; ℝ^{m}) if u ∈ L^{1}(A; ℝ^{m}), and its distributional derivative Du is a vectorvalued Radon measure on A. In this case, it turns out that S(u) is rectifiable, that is there exists a sequence (M_{i})_{i∈ℕ} of C^{1}-manifolds such that S(u) ⊆ ⋃_{i} M_{i} up to a set of H^{n−1}-measure zero; as a consequence S(u) admits a normal ν_{x} for H^{n−1}-almost every x ∈ S(u). Moreover the approximate gradient ∇u(x) exists for a.e. x ∈ A, and ∇u is the density of the absolutely continuous part of Du.

We say that u ∈ SBV (A; ℝ^{m}) if u ∈ BV (A; ℝ^{m}) and the singular part D^{s}u of its distributional derivative Du is concentrated on S(u). The space SBV (A; ℝ^{m}) is called the space of ℝ^{m}-valued special functions of bounded variation. For more details, the reader is referred to [4]. We indicate with SBV_{loc}(A, ℝ^{m}) the space of functions which belong to SBV (A′, ℝ^{m}) for every open set A′ with compact closure in A.

The set GSBV (A, ℝ^{m}) is defined as the set of functions u : A → ℝ^{m} such that φ(u) ∈ SBV_{loc}(A) for every φ ∈ C^{1}(ℝ^{m}) such that the support of ∇φ has compact closure in ℝ^{m}. If p ∈]1, +∞[, we set

GSBV^{p}(A, ℝ^{m}) := {u ∈ GSBV (A, ℝ^{m}) : ∇u ∈ L^{p}(A, M^{m×n}), H^{n−1}(S(u)) < +∞}.

By [13, Proposition 2.2] the space GSBV^{p}(A, ℝ^{m}) coincide with (GSBV^{p}(A, ℝ))^{m}, that is u := (u_{1}, …, u_{m}) ∈ GSBV^{p}(A, ℝ^{m}) if and only if u_{i} ∈ GSBV^{p}(A, ℝ) for every i = 1, …, m.

The following compactness and lower semicontinuity result will be used in the following sections. For a proof, we refer to [2].

Theorem 2.1. Let A be an open and bounded subset of ℝ^{n}. Let g(x, u) : A × ℝ^{m} → [0, ∞] be a Borel function, lower semicontinuous in u and satisfying the condition

lim_{|u|→∞} g(x, u) = +∞ for a.e. x ∈ A.

Let (u_{k})_{k∈ℕ} be a sequence in GSBV^{p}(A; ℝ^{m}) such that

lim sup_{k} ∫_{A} |∇u_{k}(x)|^{p} dx + H^{n−1} (S(u_{k})) + ∫_{A} g(x, u_{k}(x)) dx < +∞.

Then there exists a subsequence (u_{k_{h}})_{h∈ℕ} and a function u ∈ GSBV^{p}(A; ℝ^{m}) such that

(2.1)  u_{k_{h}} → u  in measure,
∇u_{k_{h}} ⇀ ∇u  weakly in L^{p}(A; M^{m×n}).

Moreover we have that

H^{n−1} (S(u)) ≤ lim inf_{h} H^{n−1} (S(u_{k_{h}})) .

Let q ∈]1, +∞[ and let us set

(2.2)  GSBV_{q}^{p}(A; ℝ^{m}) := GSBV^{p}(A; ℝ^{m}) ∩ L^{q}(A; ℝ^{m}).

We say that u_{k} ⇀ u weakly in GSBV_{q}^{p}(A; ℝ^{m}) if

u_{k} → u  in measure
(2.3)  ∇u_{k} ⇀ ∇u  weakly in L^{p}(A; M^{m×n})
u_{k} ⇀ u  weakly in L^{q}(A; ℝ^{m}).

We will often use the following fact: if u_{k} ⇀ u weakly in GSBV_{q}^{p}(A; ℝ^{m}) and Γ ⊆ A is such that H^{N−1}(Γ) < +∞ and S(u_{k}) ⊆ Γ up to a set of H^{N−1}-measure zero for all k, then S(u) ⊆ Γ up to a set of H^{N−1}-measure zero.

Γ-convergence. Let us recall the definition of De Giorgi’s Γ-convergence in metric spaces: we refer the reader to [11] for an exhaustive treatment of this subject. Let (X, d) be a metric space. We say that a sequence F_{h} : X → [−∞, +∞] Γ-converges to F : X → [−∞, +∞] (as h → +∞) if for all u ∈ X we have

(i) (Γ-liminf inequality) for every sequence (u_{h})_{h∈ℕ} converging to u in X,

lim inf_{h→+∞} F_{h}(u_{h}) ≥ F (u);

(ii) (Γ-limsup inequality) there exists a sequence (u_{h})_{h∈ℕ} converging to u in X, such that

lim sup_{h→+∞} F_{h}(u_{h}) ≤ F (u).

The function F is called the Γ-limit of (F_{h}) (with respect to d), and we write F = Γ− lim_{h} F_{h}.

We say that a family of functionals {F_{ε}} Γ-converges to F as ε → 0 if for every sequence ε_{h} → 0 as h → +∞ we have Γ− lim_{h} F_{ε_{h}} = F.

The peculiarity of this type of convergence is its variational character explained in the following proposition.

Proposition 2.2. Assume that the sequence (F_{h})_{h∈ℕ} Γ-converges to F and that there exists a compact set K ⊆ X such that for all h ∈ ℕ

inf_{u∈K} F_{h}(u) = inf_{u∈X} F_{h}(u).

Then F admits a minimum on X, inf_{X} F_{h} → min_{X} F, and any limit point of any sequence (u_{h})_{h∈ℕ} such that

lim_{h→+∞}(F_{h}(u_{h}) − inf_{u∈X} F_{h}(u)) = 0

is a minimizer of F.

Hausdorff metric on compact sets. Let A ⊆ ℝ^{2} be open and bounded, and let K(\overline{A}) be the set of all compact subsets of \overline{A}. K(\overline{A}) can be endowed by the Hausdorff metric d_{H} defined by

d_{H}(K_{1}, K_{2}) := max { sup_{x∈K_{1}} dist(x, K_{2}), sup_{y∈K_{2}} dist(y, K_{1})} ,

with the conventions dist(x, ∅) = diam(A) and sup ∅ = 0, so that d_{H}(∅, K) = 0 if K = ∅ and d_{H}(∅, K) = diam(A) if K ≠ ∅. It turns out that K(\overline{A}) endowed with the Hausdorff metric is a compact space (see e.g. [21]).

3. The quasistatic crack growth of Dal Maso-Francfort-Toader

In this section we describe the quasistatic evolution of brittle fractures proposed in [13]. They consider the case of n-dimensional finite elasticity, for an arbitrary n ≥ 1, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, depending on time. Since we are going to approximate the case n = 2, we prefer to introduce the model in this particular case. For more details, we refer the reader to [13].

Let Ω be a bounded open set of ℝ^{2} with Lipschitz boundary and let Ω_{B} be an open subset of Ω. Let ∂_{N}Ω ⊆ ∂Ω be closed in the relative topology, and let ∂_{D}Ω := ∂Ω \ ∂_{N}Ω. Let ∂_{S}Ω ⊆ ∂_{N}Ω be closed in the relative topology and such that \overline{Ω}_{B} ∩ ∂_{S}Ω = ∅. In the model proposed in [13], Ω_{B} represents the brittle part of Ω, and ∂_{D}Ω the part of the boundary on which the deformation is prescribed. Moreover the elastic body Ω is supposed to be subject to surface forces acting on ∂_{S}Ω.

Admissible cracks and deformations. The set of admissible cracks is given by

R(\overline{Ω}_{B}; ∂_{N}Ω) := \{Γ : Γ is rectifiable , Γ ˜⊂ (\overline{Ω}_{B} \ ∂_{N}Ω), H^{1}(Γ) < +∞\}.

Here A ˜⊂ B means that A ⊆ B up to a set of H^{1}-measure zero, and Γ rectifiable means that there exists a sequence (M_{i}) of C^{1}-manifolds such that Γ ˜⊂ ⋃_{i} M_{i}. If Γ is rectifiable, we can define normal vector fields ν to Γ in the following way: if Γ = ⋃_{i} Γ_{i} with Γ_{i} ˜⊂ M_{i} and Γ_{i} ∩ Γ_{j} = ∅ for i ≠ j, given x ∈ Γ_{i}, we take ν(x) = ν_{M_{i}}(x), where ν_{M_{i}}(x) is a normal vector to the C^{1}-manifold M_{i} at x. It turns out that two normal vector fields associated to different decompositions ⋃_{i} Γ_{i} of Γ coincide up to the sign H^{1} almost everywhere.

Given a crack Γ, an admissible deformation is given by any function u ∈ GSBV (Ω; ℝ^{2}) such that S(u) ˜⊂ Γ.

The surface energy. The surface energy of a crack Γ is given by

(3.1)  E^{s}(Γ) := ∫_{Γ} k(x, ν(x)) dH^{1}(x),

where ν is a unit normal vector field on Γ. Here k : \overline{Ω}_{B} × ℝ^{2} → ℝ is continuous, k(x, ·) is a norm in ℝ^{2} for all x ∈ \overline{Ω}_{B} and for all x ∈ \overline{Ω}_{B} and ν ∈ ℝ^{2}

(3.2)  K_{1}|ν| ≤ k(x, ν) ≤ K_{2}|ν|,

where K_{1}, K_{2} > 0. Notice that since k is even in the second variable, we have that the integral (3.1) is independent of the orientation given to Γ, that is independent of the particular choice of the unit normal vector field ν.

The bulk energy. Let p > 1 be fixed. Given a deformation u ∈ GSBV^{p}(Ω; ℝ^{2}) the associated bulk energy is given by

(3.3)  W(∇u) := ∫_{Ω} W (x, ∇u(x)) dx,

where W : Ω × M^{2×2} → [0, +∞) is a Carathéodory function satisfying

(3.4) for every x ∈ Ω : W (x, ·) is quasiconvex and C^{1} on M^{2×2},

(3.5) for every (x, ξ) ∈ Ω × M^{2×2} : a_{0}^{W} |ξ|^{p} − b_{0}^{W} (x) ≤ W (x, ξ) ≤ a_{1}^{W} |ξ|^{p} + b_{1}^{W} (x).

Here a_{0}^{W} , a_{1}^{W} > 0, and b_{0}^{W} , b_{1}^{W} ∈ L^{1}(Ω) are nonnegative functions. Quasiconvexity of W means that for all ξ ∈ M^{2×2} and for all φ ∈ C_{c}^{∞}(Ω; ℝ^{2})

W (ξ) ≤ ∫_{Ω} W (ξ + ∇φ) dx.

If we denote by ∂_{ξ}W : Ω × M^{2×2} → M^{2×2} the partial derivative of W with respect to ξ, it turns out that there exists a positive constant a_{2}^{W} > 0 and a nonnegative function b_{2}^{W} ∈ L^{p′}(Ω), with p′ := p/(p − 1), such that for all (x, ξ) ∈ Ω × M^{2×2}

(3.6)  |∂_{ξ}W (x, ξ)| ≤ a_{2}^{W} |ξ|^{p−1} + b_{2}^{W} (x).

By (3.5) and (3.6) the functional W, defined for all Φ ∈ L^{p}(Ω; M^{2×2}) by

W(Φ) := ∫_{Ω} W (x, Φ(x)) dx,

is of class C^{1} on L^{p}(Ω; M^{2×2}), and its differential ∂W : L^{p}(Ω; M^{2×2}) → L^{p′}(Ω; M^{2×2}) is given by

⟨∂W(Φ), Ψ⟩ = ∫_{Ω} ∂_{ξ}W (x, Φ(x))Ψ(x) dx, Φ, Ψ ∈ L^{p}(Ω; M^{2×2}),

where ⟨·, ·⟩ denotes the duality pairing between the spaces L^{p′}(Ω; M^{2×2}) and L^{p}(Ω; M^{2×2}). By (3.5) and (3.6), there exist six positive constants α_{0}^{W} > 0, α_{1}^{W} > 0, α_{2}^{W} > 0, β_{0}^{W} ≥ 0, β_{1}^{W} ≥ 0, β_{2}^{W} ≥ 0 such that for every Φ, Ψ ∈ L^{p}(Ω; M^{2×2})

α_{0}^{W}∥Φ∥_{p}^{p} − β_{0}^{W} ≤ W(Φ) ≤ α_{1}^{W}∥Φ∥_{p}^{p} + β_{1}^{W},
(3.7)  |⟨∂W(Φ), Ψ⟩| ≤ (α_{2}^{W}∥Φ∥_{p}^{p−1} + β_{2}^{W})∥Ψ∥_{p}.

The body forces. Let q > 1 be fixed. The density of applied body forces per unit volume in the reference configuration relative to the deformation u at time t ∈ [0, T] is given by ∂_{z}F (t, x, u(x)). Here F : [0, T] × Ω × ℝ^{2} → ℝ is such that:

for every z ∈ ℝ^{2} : (t, x) → F (t, x, z) is L^{1} × L^{2} measurable on [0, T] × Ω,
for every (t, x) ∈ [0, T] × Ω : z → F (t, x, z) belongs to C^{1}(ℝ^{2}),

and satisfies the following growth conditions

a_{0}^{F} |z|^{q} − b_{0}^{F} (t, x) ≤ −F (t, x, z) ≤ a_{1}^{F} |z|^{q} + b_{1}^{F} (t, x), (3.8)

|∂_{z}F (t, x, z)| ≤ a_{2}^{F} |z|^{q−1} + b_{2}^{F} (t, x)

for every (t, x, z) ∈ [0, T] × Ω × ℝ^{2}, with a_{0}^{F} > 0, a_{1}^{F} > 0 and a_{2}^{F} > 0, and where b_{0}^{F} , b_{1}^{F} ∈ C^{0}([0, T]; L^{1}(Ω)), b_{2}^{F} ∈ C^{0}([0, T]; L^{q′}(Ω)) are nonnegative functions, with q′ := q/(q − 1).

In order to deal with time variations, we assume also that for every (t, z) ∈ [0, T] × ℝ^{2}

F (t, x, z) = F (0, x, z) + ∫_{0}^{t} Ḟ (s, x, z) ds for a.e. x ∈ Ω,
∂_{z}F (t, x, z) = ∂_{z}F (0, x, z) + ∫_{0}^{t} ∂_{z}Ḟ (s, x, z) ds for a.e. x ∈ Ω,

where Ḟ : [0, T] × Ω × ℝ^{2} → ℝ is such that

for all z ∈ ℝ^{2} : (t, x) → Ḟ (t, x, z) is L^{1} × L^{2} measurable on [0, T] × Ω,
for all (t, x) ∈ [0, T] × Ω : z → Ḟ (t, x, z) is of class C^{1} on ℝ^{2},

and satisfies the growth conditions

|Ḟ (t, x, z)| ≤ a_{3}^{F} (t)|z|q̇ + b_{3}^{F} (t, x),
|∂_{z}Ḟ (t, x, z)| ≤ a_{4}^{F} (t)|z|^{q̇−1} + b_{4}^{F} (t, x)

for all (t, x, z) ∈ [0, T] × Ω × ℝ^{2}. Here 1 ≤ q̇ < q, and a_{3}^{F} , a_{4}^{F} ∈ L^{1}([0, T]), b_{3}^{F} ∈ L^{1}([0, T]; L^{1}(Ω)), b_{4}^{F} ∈ L^{1}([0, T]; L^{q̇′}(Ω)) are nonnegative functions with q̇′ := \frac{q̇}{q̇−1}.

Under the previous assumptions, for every t ∈ [0, T] the functionals

(3.9)  F(t)(u) := ∫_{Ω} F (t, x, u(x)) dx,  Ḟ(t)(u) := ∫_{Ω} Ḟ (t, x, u(x)) dx

are well defined on L^{q}(Ω; ℝ^{2}) and Lq̇(Ω; ℝ^{2}) respectively. Moreover we have that F(t) is of class C^{1} on L^{q}(Ω; ℝ^{2}), with differential ∂F(t) : L^{q}(Ω; ℝ^{2}) → L^{q′}(Ω; ℝ^{2}) defined by

⟨∂F(t)(u), v⟩ = ∫_{Ω} ∂_{z}F (t, x, u(x))v(x) dx, u, v ∈ L^{q}(Ω; ℝ^{2}),

where ⟨·, ·⟩ denotes now the duality pairing between L^{q′}(Ω; ℝ^{2}) and L^{q}(Ω; ℝ^{2}). Ḟ(t) is C^{1} on Lq̇(Ω; ℝ^{2}) with differential defined by

⟨∂Ḟ(t)(u), v⟩ = ∫_{Ω} ∂_{z}Ḟ (t, x, u(x))v(x) dx, u, v ∈ Lq̇(Ω; ℝ^{2}),

where ⟨·, ·⟩ denotes the duality pairing between L^{q̇′}(Ω; ℝ^{2}) and Lq̇(Ω; ℝ^{2}). For all u, v ∈ L^{q}(Ω; ℝ^{2}) and for all t ∈ [0, T] we have

F(t)(u) = F(0)(u) + ∫_{0}^{t} Ḟ(s)(u) ds,

(3.10)  ⟨∂F(t)(u), v⟩ = ⟨∂F(0)(u), v⟩ + ∫_{0}^{t}⟨∂Ḟ(s)(u), v⟩ ds.

Moreover we have that for every t ∈ [0, T] and for every u, v ∈ L^{q}(Ω; ℝ^{n})

α_{0}^{F}∥u∥_{q}^{q} − β_{0}^{F} ≤ −F(t)(u) ≤ α_{1}^{F}∥u∥_{q}^{q} + β_{1}^{F},

|⟨∂F(t)(u), v⟩| ≤ (α_{2}^{F}∥u∥_{q}^{q−1} + β_{2}^{F})∥v∥_{q}, (3.11)

|Ḟ(t)(u)| ≤ α_{3}^{F}(t)∥u∥q̇ ; q̇ + β_{3}^{F}(t), (3.12)

|⟨∂Ḟ(t)(u), v⟩| ≤ (α_{4}^{F}(t)∥u∥q̇−1 ; q̇ + β_{4}^{F}(t))∥v∥q̇, (3.13)

where α_{0}^{F} > 0, α_{1}^{F} > 0, α_{2}^{F} > 0, β_{0}^{F} ≥ 0, β_{1}^{F} ≥ 0, β_{2}^{F} ≥ 0 are positive constants, and α_{3}^{F}, α_{4}^{F}, β_{3}^{F}, β_{4}^{F} ∈ L^{1}([0, T]) are nonnegative functions.

The surface forces. The density of the surface forces on ∂_{S}Ω at time t under the deformation u is given by ∂_{z}G(t, x, u(x)), where G : [0, T] × ∂_{S}Ω × ℝ^{2} → ℝ is such that

for every z ∈ ℝ^{2} : (t, x) → G(t, x, z) is L^{1} × H^{1}-measurable,
for every (t, x) ∈ [0, T] × ∂_{S}Ω : z → G(t, x, z) belongs to C^{1}(ℝ^{2}),

and satisfies the growth conditions

− a_{0}^{G}(t, x)|z| − b_{0}^{G}(t, x) ≤ −G(t, x, z) ≤ a_{1}^{G}|z|^{r} + b_{1}^{G}(t, x),
|∂_{z}G(t, x, z)| ≤ a_{2}^{G}|z|^{r−1} + b_{2}^{G}(t, x),

for every (t, x, z) ∈ [0, T] × ∂_{S}Ω × ℝ^{2}. Here r is an exponent related to the trace operators on Sobolev spaces: if p < 2, then we suppose that p ≤ r ≤ \frac{p}{2−p}, while if p ≥ 2, we suppose p ≤ r. Moreover a_{1}^{G} ≥ 0, a_{2}^{G} ≥ 0 are two nonnegative constants, and a_{0}^{G} ∈ L^{∞}([0, T]; L^{r′}(∂_{S}Ω)), b_{0}^{G}, b_{1}^{G} ∈ C^{0}([0, T]; L^{1}(∂_{S}Ω)), and b_{2}^{G} ∈ C^{0}([0, T]; L^{r′}(∂_{S}Ω)) are nonnegative functions with r′ := r/(r − 1)

We assume that for every (t, z) ∈ [0, T] × ℝ^{2}

G(t, x, z) = G(0, x, z) + ∫_{0}^{t} Ġ(s, x, z)ds for H^{1}-a.e. x ∈ ∂_{S}Ω,
∂_{z}G(t, x, z) = ∂_{z}G(0, x, z) + ∫_{0}^{t} ∂_{z}Ġ(s, x, z)ds for H^{1}-a.e. x ∈ ∂_{S}Ω,

where Ġ : [0, T] × ∂_{S}Ω × ℝ^{2} → ℝ is such that

for all z ∈ ℝ^{2} : (t, x) → Ġ(t, x, z) is L^{1} × H^{1}-measurable,
for all (t, x) ∈ [0, T] × ∂_{S}Ω : z → Ġ(t, x, z) belongs to C^{1}(ℝ^{2}),

and satisfies the the growth conditions

|Ġ(t, x, z)| ≤ a_{3}^{G}(t)|z|^{r} + b_{3}^{G}(t, x),
|∂_{z}Ġ(t, x, z)| ≤ a_{4}^{G}(t)|z|^{r−1} + b_{4}^{G}(t, x)

for all (t, x, z) ∈ [0, T] × ∂_{S}Ω × ℝ^{2}. Here a_{3}^{G}, a_{4}^{G} ∈ L^{1}([0, T]), b_{3}^{G} ∈ L^{1}([0, T]; L^{1}(∂_{S}Ω)) and b_{4}^{G} ∈ L^{1}([0, T]; L^{r′}(∂_{S}Ω)) are nonnegative functions.

By the previous assumptions, the following functionals on L^{r}(∂_{S}Ω; ℝ^{2})

(3.14)  G(t)(u) := ∫_{∂_{S}Ω} G(t, x, u(x)) dH^{1}(x), Ġ(t)(u) := ∫_{∂_{S}Ω} Ġ(t, x, u(x)) dH^{1}(x)

are well defined. For every t ∈ [0, T] we have that G(t) is of class C^{1} on L^{r}(∂_{S}Ω; ℝ^{2}) and its differential is given by

⟨∂G(t)(u), v⟩ = ∫_{∂_{S}Ω} ∂_{z}G(t, x, u(x))v(x) dH^{1}(x), u, v ∈ L^{r}(∂_{S}Ω; ℝ^{2}),

where ⟨·, ·⟩ denotes now the duality pairing between L^{r′}(∂_{S}Ω; ℝ^{2}) and L^{r}(∂_{S}Ω; ℝ^{2}). Moreover, Ġ(t) is of class C^{1} on L^{r}(∂_{S}Ω; ℝ^{2}), and its differential is given by

⟨∂Ġ(t)(u), v⟩ = ∫_{∂_{S}Ω} ∂_{z}Ġ(t, x, u(x))v(x) dH^{1}(x)

for all u, v ∈ L^{r}(∂_{S}Ω; ℝ^{2}). Finally we have

G(t)(u) = G(0)(u) + ∫_{0}^{t} Ġ(s)(u) ds,

⟨∂G(t)(u), v⟩ = ⟨∂G(0)(u), v⟩ + ∫_{0}^{t}⟨∂Ġ(s)(u), v⟩ ds,

for every u, v ∈ L^{r}(∂_{S}Ω; ℝ^{2}).

Let Ω_{S} ⊆ Ω \ \overline{Ω}_{B} be open with Lipschitz boundary, and such that ∂_{S}Ω ⊆ ∂Ω_{S}; the trace operator from W^{1,p}(Ω_{S}; ℝ^{2}) into L^{r}(∂Ω_{S}; ℝ^{2}) is then compact, and so there exists a constant γ_{S} > 0 such that

(3.15)  ∥u∥_{r,∂_{S}Ω} ≤ γ_{S}(∥∇u∥_{p,Ω_{S}} + ∥u∥_{p,Ω_{S}})

for every u ∈ W^{1,p}(Ω_{S}; ℝ^{2}). By the previous assumptions, we have that there exist six nonnegative constants α_{0}^{G}, α_{1}^{G}, α_{2}^{G}, β_{0}^{G}, β_{1}^{G}, β_{2}^{G} and four nonnegative functions α_{3}^{G}, α_{4}^{G}, β_{3}^{G}, β_{4}^{G} ∈ L^{1}([0, T]), such that

− α_{0}^{G}∥u∥_{r,∂_{S}Ω} − β_{0}^{G} ≤ −G(t)(u) ≤ α_{1}^{G}∥u∥_{r,∂_{S}Ω}^{r} + β_{1}^{G},

|⟨∂G(t)(u), v⟩| ≤ (α_{2}^{G}∥u∥_{r,∂_{S}Ω}^{r−1} + β_{2}^{G})∥v∥_{r,∂_{S}Ω}, (3.16)

|Ġ(t)(u)| ≤ α_{3}^{G}(t)∥u∥_{r,∂_{S}Ω}^{r} + β_{3}^{G}(t), (3.17)

|⟨∂Ġ(t)(u), v⟩| ≤ (α_{4}^{G}(t)∥u∥_{r,∂_{S}Ω}^{r−1} + β_{4}^{G}(t))∥v∥_{r,∂_{S}Ω}

for every t ∈ [0, T] and u, v ∈ L^{r}(∂_{S}Ω; ℝ^{2}).

Configurations with finite energy. The deformations on the boundary ∂_{D}Ω are given by (the traces of) functions g ∈ W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}), where p, q are the exponents in (3.5) and (3.8) respectively. Given a crack Γ ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and a boundary deformation g, the set of admissible deformations with finite energy relative to (g, Γ) is defined by

AD(g, Γ) := {u ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) : S(u) ˜⊂ Γ, u = g H^{1}-a.e. on ∂_{D}Ω \ Γ},

where we recall that

GSBV_{q}^{p}(Ω; ℝ^{2}) := GSBV^{p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}),

and the equality u = g on ∂_{D}Ω \ Γ is intended in the sense of traces (see [13, Section 2]).

Note that if u ∈ GSBV_{q}^{p}(Ω; ℝ^{2}), then W(u) < +∞ and |F(t)(u)| < +∞ for all t ∈ [0, T]. Moreover since Γ ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and S(u) ˜⊂ Γ, we have that u ∈ W^{1,p}(Ω_{S}; ℝ^{2}) ∩ L^{q}(Ω_{S}; ℝ^{2}) so that G(t)(u) is well defined and |G(t)(u)| < +∞ for all t ∈ [0, T]. Notice that there exists always a deformation without crack which satisfies the boundary condition, namely the function g itself.

The total energy. For every t ∈ [0, T], the total energy relative to the configuration (u, Γ) with u ∈ AD(g, Γ) is given by

(3.18)  E(t)(u, Γ) := E^{el}(t)(u) + E^{s}(Γ),

where

(3.19)  E^{el}(t)(u) := W(u) − F(t)(u) − G(t)(u),

and W, F(t), G(t) and E^{s} are defined in (3.3), (3.9), (3.14) and (3.1) respectively. It turns out that there exist four constants α_{0}^{E} > 0, α_{1}^{E} > 0, β_{0}^{E} ≥ 0, β_{1}^{E} ≥ 0 such that

(3.20)  E^{el}(t)(u) ≥ α_{0}^{E}(∥∇u∥_{p}^{p} + ∥u∥_{q}^{q}) − β_{0}^{E},
E^{el}(t)(u) ≤ α_{1}^{E}(∥∇u∥_{p}^{p} + ∥u∥_{q}^{q} + ∥u∥_{r,∂_{S}Ω}^{r}) + β_{1}^{E},

for every t ∈ [0, T] and u ∈ GSBV_{q}^{p}(Ω; ℝ^{2}).

The time dependent boundary deformations. We will consider boundary deformations g(t) such that

t → g(t) ∈ AC([0, T]; W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})),

so that

t → ġ(t) ∈ L^{1}([0, T]; W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})),

and

t → ∇ġ(t) ∈ L^{1}([0, T]; L^{p}(Ω; M^{2×2})).

The existence result. Let Γ_{0} ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) be a preexisting crack. The next Theorem proved in [13] establishes the existence of a quasistatic evolution with preexisting crack Γ_{0}.

Theorem 3.1. Let Γ_{0} ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) be a preexisting crack. Then there exists a quasistatic evolution with preexisting crack Γ_{0} and boundary deformation g(t), i.e., there exists a function t → (u(t), Γ(t)) from [0, T] to GSBV_{q}^{p}(Ω; ℝ^{2}) × R(\overline{Ω}_{B}; ∂_{N}Ω) with the following properties:

(a) (u(0), Γ(0)) is such that

E(0)(u(0), Γ(0)) = min{E(0)(v, Γ) : v ∈ AD(g(0), Γ), Γ_{0} ˜⊂ Γ};

(b) u(t) ∈ AD(g(t), Γ(t)) for all t ∈ [0, T];

(c) irreversibility: Γ_{0} ˜⊂ Γ(s) ˜⊂Γ(t) whenever 0 ≤ s < t ≤ T;

(d) static equilibrium: for all t ∈ [0, T]

E(t)(u(t), Γ(t)) = min{E(t)(v, Γ) : v ∈ AD(g(t), Γ), Γ(t) ˜⊂ Γ};

(e) nondissipativity: the function t → E(t) := E(t)(u(t), Γ(t)) is absolutely continuous on [0, T], and for a.e. t ∈ [0, T]

Ė(t) = ⟨∂W(∇u(t)), ∇ġ(t)⟩ − ⟨∂F(t)(u(t)), ġ(t)⟩ − Ḟ(t)(u(t)) (3.21)

−⟨∂G(t)(u(t)), ġ(t)⟩ − Ġ(t)(u(t)).

The next theorem gives a compactness and lower semicontinuity result with respect to weak convergence in GSBV_{q}^{p}(Ω, ℝ^{2}) which will be often used in the next sections.

Theorem 3.2. Let t_{k} ∈ [0, T] with t_{k} → t, and let (u_{k}) ⊂ GSBV_{q}^{p}(Ω; ℝ^{2}), C ∈]0; +∞[ such that S(u_{k}) ˜⊂ \overline{Ω}_{B} and

E^{el}(t_{k})(u_{k}) + E^{s}(S(u_{k})) ≤ C,

where E^{el} and E^{s} are defined as in (3.19) and (3.1). Then there exists a subsequence (u_{k_{h}})_{h∈ℕ} converging to some u weakly in GSBV_{q}^{p}(Ω; ℝ^{2}) such that S(u) ˜⊂ \overline{Ω}_{B},

E^{el}(t)(u) ≤ lim inf_{h→∞} E^{el}(t_{k_{h}})(u_{k_{h}}),

and

E^{s}(S(u)) ≤ lim inf_{h→∞} E^{s}(S(u_{k_{h}})).

Proof. By (3.20) and (3.2), we have that there exists C′ ∈]0, +∞[ such that

∥∇u_{k}∥_{p}^{p} + ∥u_{k}∥_{q}^{q} + H^{1}(S(u_{k})) ≤ C′.

Then we can apply Theorem 2.1 with g(x, u_{k}) = |u_{k}|^{q}, obtaining a subsequence (u_{k_{h}})_{h∈ℕ} and u ∈ GSBV^{p}(Ω; ℝ^{2}) such that (2.1) holds: in particular we may assume that u_{k_{h}} → u pointwise a.e.. We have u_{k_{h}} → u strongly in L^{1}(Ω; ℝ^{2}), and by Fatou’s Lemma we have that u ∈ L^{q}(Ω; ℝ^{2}) so that u ∈ GSBV_{q}^{p}(Ω; ℝ^{2}). We conclude u_{k_{h}} ⇀ u weakly in GSBV_{q}^{p}(Ω; ℝ^{2}). By [2, Theorem 3.7] we have that

E^{s}(S(u)) ≤ lim inf_{h} E^{s}(S(u_{k_{h}})),

by [18] we have that

∫_{Ω} W (x, ∇u) dx ≤ lim inf_{h} ∫_{Ω} W (x, ∇u_{k_{h}}) dx,

and by Fatou’s Lemma (in the limsup version) we have

lim sup_{h} ∫_{Ω} F (t_{k_{h}}, x, u_{k_{h}}(x)) dx ≤ ∫_{Ω} F (t, x, u(x)) dx.

Since (u_{h_{k}})_{|Ω_{S}} is bounded in W^{1,p}(Ω_{S}; ℝ^{2})∩L^{q}(Ω_{S}; ℝ^{2}), and the trace operator from W^{1,p}(Ω_{S}; ℝ^{2}) into L^{r}(Ω_{S}; ℝ^{2}) is compact, we get

lim_{h} G(t_{k_{h}})(u_{k_{h}}) = G(t)(u),

and so the proof is thus concluded. □

Adaptive vertex  Adaptive edge  Adaptive triangulation

Figure 1

4. The finite element space and the transfer of jump

Let Ω ⊆ ℝ^{2} be a polygonal set and let us fix two positive constants 0 < c_{1} < c_{2} < +∞. By a regular triangulation of Ω of size ε we intend a finite family of (closed) triangles T_{i} such that \overline{Ω} = ⋃_{i} T_{i}, T_{i} ∩ T_{j} is either empty or equal to a common edge or to a common vertex, and each T_{i} contains a ball of diameter c_{1}ε and is contained in a ball of diameter c_{2}ε.

We indicate by R_{ε}(Ω) the family of all regular triangulations of Ω of size ε. It turns out that there exist 0 < ϑ_{1} < ϑ_{2} < π such that for all triangle T belonging to a regular triangulation T ∈ R_{ε}(Ω), the inner angles of T are between ϑ_{1} and ϑ_{2}. Moreover, every edge of T has length greater than c_{1}ε and lower than c_{2}ε.

Let ε > 0, R_{ε} ∈ R_{ε}(Ω), and let a ∈]0, \frac{1}{2}[. Let us consider a triangulation T nested in R_{ε} obtained dividing each triangle T ∈ R_{ε} into four triangles taking over every edge [x, y] of T a knot z which satisfies

z = tx + (1 − t)y, t ∈ [a, 1 − a].

We will call these vertices adaptive, the triangles obtained gluing these points adaptive triangles, and their edges adaptive edges. We denote by T_{ε,a}(Ω) the set of triangulations T constructed in this way. Note that for all T ∈ T_{ε,a}(Ω) there exist 0 < c_{1}^{a} < c_{2}^{a} < +∞ such that every T_{i} ∈ T contains a ball of diameter c_{1}^{a}ε and is contained in a ball of diameter c_{2}^{a}ε. Then there exist 0 < ϑ_{1}^{a} < ϑ_{2}^{a} < π such that for all T belonging to T ∈ T_{ε,a}(Ω), the inner angles of T are between ϑ_{1}^{a} and ϑ_{2}^{a}. Moreover, every edge of T has length greater than c_{1}^{a}ε and lower than c_{2}^{a}ε.

From now on for all ε > 0 we fix R_{ε} ∈ R_{ε}(Ω). We suppose that the brittle part Ω_{B} and the region Ω_{S} introduced before for the model of quasistatic growth of fractures are composed of triangles of R_{ε} for all ε. Moreover we suppose that ∂_{D}Ω and ∂_{S}Ω are composed of edges of R_{ε} for all ε up to a finite number of points. Finally, in order to deal with the deformation at the boundary, it will be useful to consider Ω_{D} polygonal such that Ω_{D} ∩ Ω = ∅, and ∂Ω_{D} ∩ ∂Ω = ∂_{D}Ω up to a finite number of points. We set

(4.1)  Ω′ := Ω ∪ Ω_{D} ∪ ∂_{D}Ω,

and we suppose that the regular triangulation R_{ε} can be extended to a regular triangulation of Ω′, so that every triangulation T in T_{ε,a}(Ω) can be extended to a triangulation of T_{ε,a}(Ω′) considering the middle points of the new edges as adaptive vertices: we indicate these extended triangulation with the same symbol T.

We consider the following discontinuous finite element space. We indicate by A_{ε,a}(Ω; ℝ^{2}) the set of all u : Ω → ℝ^{2} such that there exists a triangulation T(u) ∈ T_{ε,a}(Ω) nested in R_{ε} with u affine on every triangle T ∈ T(u). For every u ∈ A_{ε,a}(Ω; ℝ^{2}), we indicate by S(u) the family of edges of T(u) inside Ω across which u is discontinuous. Notice that u ∈ SBV (Ω; ℝ^{2}) and that the notation is consistent with the usual one employed in the theory of functions of bounded variation. Let us set

(4.2)  AF_{ε}(Ω; ℝ^{2}) := {u : Ω → ℝ^{2} : u is continuous and affine on each triangle T ∈ R_{ε}}.

P π^{a}(P)  The interpolating curve S_{ε,a} The projection π_{a}(P)  S  S_{ε,a}

Figure 2

The discretization of the problem will be carried out using the space

(4.3)  A_{ε,a}^{B}(Ω; ℝ^{2}) := \{u ∈ A_{ε,a}(Ω; ℝ^{2}) : S(u) ⊆ \overline{Ω}_{B}\}.

Given any g ∈ AF_{ε}(Ω; ℝ^{2}), for every u ∈ A_{ε,a}^{B}(Ω; ℝ^{2}) let

(4.4)  S_{D}^{g}(u) := {x ∈ ∂_{D}Ω : u(x) ≠ g(x)},

that is S_{D}^{g}(u) denotes the set of edges of ∂_{D}Ω at which the boundary condition is not satisfied. For every u ∈ A_{ε,a}^{B}(Ω; ℝ^{2}), let us also set

(4.5)  S^{g}(u) := S(u) ∪ S_{D}^{g}(u).

An essential tool in the approximation result of this paper is Proposition 4.2 which generalizes the piecewise affine transfer of jump [17, Proposition 5.1] to the case of vector valued functions with bulk energy E^{el} and surface energy E^{s} of the form (3.19) and (3.1) respectively.

In order to deal with the surface energy E^{s} we will need the following geometric construction. Let S ⊆ Ω be a segment and let us suppose that S intersects the edges of R_{ε} at most in one point for all ε > 0. Let a ∈]0, \frac{1}{2}[, and let P = S ∩ ζ, where ζ = [x, y] is an edge of R_{ε}: we indicate with π_{a}(P) the projection of P on the segment {tx + (1 − t)y : t ∈ [a, 1 − a]}. The interpolating curve S_{ε,a} of S in R_{ε} with parameter a is given connecting all the π_{a}(P)’s belonging to the same triangle of R_{ε} (see Figure 2).

Lemma 4.1. Under the previous assumptions, there exists a function η(a) independent of S with η(a) → 0 as a → 0 such that

lim sup_{ε→0} |E^{s}(S_{ε,a}) − E^{s}(S)| ≤ η(a)E^{s}(S),

where E^{s} is defined in (3.1).

Proof. By (3.2), we have that there exist ω and K_{3} > 0 such that for all x_{1}, x_{2} ∈ \overline{Ω} and |ν_{1}| = |ν_{2}| = 1

|k(x_{1}, ν_{1}) − k(x_{2}, ν_{2})| ≤ ω(|x_{1} − x_{2}|) + K_{3}|ν_{1} − ν_{2}|,

where ω : ]0, +∞[→]0, +∞[ is a decreasing function such that ω(s) → 0 as s → 0. Let T ∈ R_{ε} be such that T ∩ S ≠ ∅, and let us choose x_{T} ∈ T ∩ S and x_{T}^{ε,a} ∈ T ∩ S_{ε,a}. Let c_{2} > 0 denote the characteristic constant of R_{ε} such that every T ∈ R_{ε} is contained in a ball of diameter c_{2}ε. Then we have

|∫_{S_{ε,a}∩T} k(x, ν_{T}^{ε,a}) dH^{1} − ∫_{S∩T} k(x, ν_{T}) dH^{1}|

≤ |∫_{S_{ε,a}∩T} k(x_{T}^{ε,a}, ν_{T}^{ε,a}) dH^{1} − ∫_{S∩T} k(x_{T} , ν_{T}) dH^{1}| + ω(c_{2}ε)H^{1}(S_{ε,a} ∩ T) + ω(c_{2}ε)H^{1}(S ∩ T)

≤ |k(x_{T}^{ε,a}, ν_{T}^{ε,a})H^{1}(S_{ε,a} ∩ T) − k(x_{T} , ν_{T})H^{1}(S ∩ T)| + ω(c_{2}ε) [H^{1}(S_{ε,a} ∩ T) + H^{1}(S ∩ T)] , where ν_{T}^{ε,a}, ν_{T} are the (constant) normal to S_{ε,a} ∩ T and S ∩ T respectively. We have

|k(x_{T}^{ε,a}, ν_{T}^{ε,a})H^{1}(S_{ε,a} ∩ T) − k(x_{T} , ν_{T})H^{1}(S ∩ T)|

≤ k(x_{T}^{ε,a}, ν_{T}^{ε,a}) |H^{1}(S_{ε,a} ∩ T) − H^{1}(S ∩ T)| + |k(x_{T}^{ε,a}, ν_{T}^{ε,a}) − k(x_{T} , ν_{T})| H^{1}(S ∩ T)

≤ K_{2} |H^{1}(S_{ε,a} ∩ T) − H^{1}(S ∩ T)| + ω(|x_{T}^{ε,a} − x_{T} |)H^{1}(S ∩ T) + K_{3}|ν_{T}^{ε,a} − ν_{T} |H^{1}(S ∩ T),

where K_{2} is defined in (3.2). We are now ready to conclude: in fact, following [19, Lemma 5.2.2], we can choose the orientation of ν_{T}^{ε,a} in such a way that

|ν_{T}^{ε,a} − ν_{T} |H^{1}(S ∩ T) ≤ D_{2}aε,  |H^{1}(S_{ε,a} ∩ T) − H^{1}(S ∩ T)| ≤ D_{1}aε,

with D_{1}, D_{2} > 0 independent of T, ε, a. Then, summing up the preceding inequalities, recalling that the number of triangles of R_{ε} intersecting S is less than c̃ε^{−1}H^{1}(S) for ε small enough, with c̃ independent of S and ε (see for example [17, Lemma 2.5]), we obtain

lim sup_{ε→0} |E^{s}(S_{ε,a}) − E^{s}(S)| ≤ ρ(a)H^{1}(S),

where ρ(a) := c̃(K_{2}D_{1} + K_{3}D_{2})a. In view of (3.2), we conclude that

lim sup_{ε→0} |E^{s}(S_{ε,a}) − E^{s}(S)| ≤ K_{1}^{−1}ρ(a)E^{s}(S),

and so the proof is concluded choosing η(a) := K_{1}^{−1}ρ(a). □

For all u ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) and for all g ∈ W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}), let us set

(4.6)  S^{g}(u) := S(u) ∪ {x ∈ ∂_{D}Ω : u(x) ≠ g(x)},

where the inequality is intended in the sense of traces. We are now in a position to state the piecewise affine transfer of jump proposition in our setting.

Proposition 4.2. Let a ∈]0, \frac{1}{2}[, and for all i = 1, …, m let

u_{ε}^{i} ∈ A_{ε,a}^{B}(Ω; ℝ^{2}), u^{i} ∈ GSBV_{q}^{p}(Ω; ℝ^{2})

be such that

u_{ε}^{i} ⇀ u^{i} weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

Let moreover g_{ε}^{i}, h_{ε} ∈ AF_{ε}(Ω; ℝ^{2}), g^{i}, h ∈ W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}) be such that

g_{ε}^{i} → g^{i}, h_{ε} → h strongly in W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}).

Then for every v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, there exists v_{ε} ∈ A_{ε,a}^{B}(Ω; ℝ^{2}) such that

∇v_{ε} → ∇v strongly in L^{p}(Ω; M^{2×2}),
v_{ε} → v strongly in L^{q}(Ω; ℝ^{2}),

and such that

lim sup_{ε→0} E^{s} (S^{h_{ε}}(v_{ε}) \ ⋃_{i=1}^{m} S^{g_{ε}^{i}}(u_{ε}^{i})) ≤ μ(a)E^{s} (S^{h}(v) \ ⋃_{i=1}^{m} S^{g^{i}}(u^{i})) ,

where μ(a) depends only on a, μ(a) → 1 as a → 0, and E^{s} is defined in (3.1). In particular for all t ∈ [0, T] and for all t_{ε} → t we have

E^{el}(t_{ε})(v_{ε}) → E^{el}(t)(v),

where E^{el} is defined in (3.19).

The proof of Proposition 4.2 can be obtained from that of [17, Proposition 5.1] taking into account the following modifications. We can consider v scalar valued since vector valued maps can be easily dealt componentwise. Even if the surface energy is of the form (3.1), we can still restrict ourselves to the case in which v has piecewise linear jumps outside a suitable neighborhood of ⋃_{i=1}^{m} S^{g^{i}}(u^{i}) by using the density result of [10]. In order to approximate the piecewise linear jumps, we use Lemma 4.1. Finally the fact that p ≠ 2 prevents us from considering the piecewise jumps as union of disjoint segments: we overcome this difficulty choosing v_{ε} = 0 in the regular triangles which contain the intersection points, and then interpolating v outside as in [17, Proposition 5.1].

5. Preexisting cracks and their approximation

In Section 7, we will need to approximate the surface energy of a given preexisting crack Γ^{0}. We take the initial crack in the class

(5.1) Γ(Ω) := \{Γ ˜⊂ \overline{Ω}_{B} : H^{1}(Γ) < +∞, Γ = S^{h}(z)

for some h ∈ W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}) and z ∈ GSBV_{q}^{p}(Ω; ℝ^{2})}.

Notice that it is not restrictive to assume h ≡ 0. We take as discretization of Γ(Ω) the following class

(5.2)  Γ_{ε,a}(Ω) := \{Γ ˜⊂ \overline{Ω}_{B} : H^{1}(Γ) < +∞, Γ = S^{0}(z) for some z ∈ A_{ε,a}^{B}(Ω; ℝ^{2})\}.

We have the following approximation result.

Proposition 5.1. Let Γ^{0} ∈ Γ(Ω). Then for every ε > 0 and a ∈]0, \frac{1}{2}[ there exists Γ_{ε,a}^{0} ∈ Γ_{ε,a}(Ω) such that

lim_{ε,a→0} E^{s}(Γ_{ε,a}^{0}) = E^{s}(Γ^{0}),

where E^{s} is defined in (3.1).

Moreover let g_{ε} ∈ AF_{ε}(Ω; ℝ^{2}), g ∈ W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}) be such that as ε → 0

g_{ε} → g strongly in W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2}),

and let us consider

F_{ε,a}(v) := \begin{cases}E^{el}(0)(v) + E^{s} (S^{g_{ε}}(v) \ Γ_{ε,a}^{0}) & if v ∈ A_{ε,a}^{B}(Ω; ℝ^{2}), \\ +∞ & otherwise in L^{1}(Ω; ℝ^{2}), \end{cases}

and

F (v) := \begin{cases}E^{el}(0)(v) + E^{s} (S^{g}(v) \ Γ^{0}) & if v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}), S(v) ˜⊂ \overline{Ω}_{B}, \\ +∞ & otherwise in L^{1}(Ω; ℝ^{2}), \end{cases}

where E^{el} is defined in (3.19). Then the family (F_{ε,a}) Γ-converges to F in the strong topology of L^{1}(Ω; ℝ^{2}) as ε → 0 and a → 0.

Proof. Let us consider Γ^{0} ∈ Γ(Ω) with Γ^{0} = S^{0}(z) for some z ∈ GSBV_{q}^{p}(Ω; ℝ^{2}). Then by Proposition 4.2 for every ε > 0 and a ∈ (0, \frac{1}{2}), there exists z̃_{ε,a} ∈ A_{ε,a}(Ω; ℝ^{2}) such that for ε → 0 and for all a

∇z̃_{ε,a} → ∇z strongly in L^{p}(Ω; M^{2×2}),
z̃_{ε,a} → z strongly in L^{q}(Ω; ℝ^{2}),

and

lim sup_{ε→0} E^{s}(S^{0}(z̃_{ε,a})) ≤ μ(a)E^{s}(S^{0}(z))

with μ(a) → 1 as a → 0, where E^{s} is defined in (3.1). Let a_{i} ↘ 0, and let ε_{i} ↘ 0 be such that for all ε ≤ ε_{i}

E^{s}(S^{0}(z̃_{ε,a_{i}})) ≤ μ(a_{i})E^{s}(S^{0}(z)) + a_{i},

and

∥∇z̃_{ε,a_{i}} − ∇z∥_{L^{p}(Ω;M^{2×2})} ≤ a_{i}, ∥z̃_{ε,a_{i}} − z∥_{L^{q}(Ω;ℝ^{2})} ≤ a_{i}.

Setting

z_{ε,a} := \begin{cases}z̃_{ε,a_{i}} & ε_{i+1} < ε ≤ ε_{i}, a ≤ a_{i}, \\ z̃_{ε,a_{j−1}} & ε_{i+1} < ε ≤ ε_{i}, a_{j} < a ≤ a_{j−1}, j ≤ i, \end{cases}

we have that

lim_{ε,a→0} ∇z_{ε,a} = ∇z strongly in L^{p}(Ω; M^{2×2}),

lim_{ε,a→0} z_{ε,a} = z strongly in L^{q}(Ω; ℝ^{2}),

and

lim sup_{ε,a→0} E^{s}(S^{0}(z_{ε,a})) ≤ E^{s}(S^{0}(z)).

Since by Theorem 3.2 we have E^{s}(S^{0}(z_{ε,a})) ≤ lim inf_{ε,a→0} E^{s}(S^{0}(z_{ε,a})), we conclude that

lim_{ε,a→0} E^{s}(S^{0}(z_{ε,a})) = E^{s}(S^{0}(z)).

Let us set for every ε, a

Γ_{ε,a}^{0} := S^{0}(z_{ε,a}).

We have that

lim_{ε,a→0} E^{s}(Γ_{ε,a}^{0}) = E^{s}(Γ^{0}).

Let us come to the second part of the proof. Let us consider (ε_{n}, a_{n})_{n∈ℕ} such that ε_{n} → 0 and a_{n} → 0. If we prove that (F_{ε_{n},a_{n}})_{n∈ℕ} Γ-converges to F in the strong topology of L^{1}(Ω; ℝ^{2}), the proposition is proved since the sequence is arbitrary. Since we can reason up to subsequences, it is not restrictive to assume a_{n} ↘ 0.

Let us start with the Γ-limsup inequality considering v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}), with S(v) ⊆ \overline{Ω}_{B}. For any n fixed, by Proposition 4.2 there exists ṽ_{ε,a_{n}} ∈ A_{ε,a_{n}}^{B}(Ω; ℝ^{2}) such that for ε → 0

∇ṽ_{ε,a_{n}} → ∇v strongly in L^{p}(Ω; M^{2×2}),
ṽ_{ε,a_{n}} → v strongly in L^{q}(Ω; ℝ^{2}),

and such that

lim sup_{ε→0} E^{s}(S^{g_{ε}}(ṽ_{ε,a_{n}}) \ Γ_{ε,a_{n}}^{0}) ≤ μ(a_{n})E^{s}(S^{g_{ε}}(v) \ Γ^{0})

with μ(a) → 1 as a → 0. For every m ∈ ℕ let ε^{m} be such that for all ε ≤ ε^{m}

E^{s}(S^{g_{ε}}(ṽ_{ε,a_{m}}) \ Γ_{ε,a_{m}}^{0}) ≤ μ(a_{m})E^{s}(S^{g}(v) \ Γ^{0}) + a_{m},

and

∥∇ṽ_{ε,a_{m}} − ∇v∥_{L^{p}(Ω;M^{2×2})} ≤ a_{m}, ∥ṽ_{ε,a_{m}} − v∥_{L^{q}(Ω;ℝ^{2})} ≤ a_{m}.

We can assume ε^{m} ↘ 0. Setting

v_{ε_{n},a_{n}} := \begin{cases}ṽ_{ε_{n},a_{m}} & ε^{m+1} < ε_{n} ≤ ε^{m}, n ≥ m, \\ ṽ_{ε_{n},a_{n}} & ε^{m+1} < ε_{n} ≤ ε^{m}, n < m, \end{cases}

we have that

lim_{n} ∇v_{ε_{n},a_{n}} = ∇v strongly in L^{p}(Ω; M^{2×2}),
lim_{n} v_{ε_{n},a_{n}} = v strongly in L^{q}(Ω; ℝ^{2}),

and

lim sup_{n} E^{s}(S^{g_{ε}}(v_{ε_{n},a_{n}}) \ Γ_{ε,a}^{0}) ≤ E^{s}(S^{g}(v) \ Γ^{0}).

Then we get

lim sup_{n} F_{ε_{n},a_{n}}(v_{ε_{n},a_{n}}) ≤ lim sup_{n} E^{el}(0)(v_{ε_{n},a_{n}}) + lim sup_{n} E^{s}(S^{g_{εn}} (v_{ε_{n},a_{n}}) \ Γ_{ε_{n},a_{n}}^{0})
≤ E^{el}(0)(v) + E^{s}(S^{g}(v) \ Γ^{0}) = F (v),

so that the Γ-limsup inequality holds.

Let us come to the Γ-liminf inequality. Let v_{n}, v ∈ L^{1}(Ω; ℝ^{2}) be such that v_{n} → v strongly in L^{1}(Ω; ℝ^{2}) and lim inf_{n} F_{ε_{n},a_{n}}(v_{n}) < +∞. By Theorem 3.2, we have v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B} and

E^{el}(0)(v) ≤ lim inf_{n} E^{el}(0)(v_{n}).

Let us consider Ω′ defined in (4.1). Let us extend g_{ε_{n}} and g to W^{1,p}(Ω′; ℝ^{2}) ∩ L^{q}(Ω′; ℝ^{2}) in such a way that g_{ε_{n}} → g strongly in W^{1,p}(Ω′; ℝ^{2}) ∩ L^{q}(Ω′; ℝ^{2}), and let us also extend v_{n}, v to Ω′ setting v_{n} = g_{ε_{n}} and v = g on Ω_{D}. We indicate these extensions with w_{n} and w respectively. Let us also set z_{ε_{n},a_{n}} = z = 0 on Ω_{D}, where z_{ε_{n},a_{n}} and z are such that Γ_{ε,a}^{0} = z_{ε_{n},a_{n}} and Γ^{0} = S(z). We indicate these extension by ζ_{ε_{n},a_{n}} and ζ respectively. Then for every η > 0 we have by Theorem 3.2

E^{s}(S(w + ηζ)) ≤ lim inf_{n} E^{s}(S(w_{n} + ηζ_{ε_{n},a_{n}})).

Since for a.e. η > 0 we have S(w + ηζ) = S(w) ∪ S(ζ) and S(w_{n} + ηζ_{ε_{n},a_{n}}) = S(w_{n}) ∪ S(ζ_{ε_{n},a_{n}}), we deduce that

E^{s}(S^{g}(v) ∪ Γ^{0}) ≤ lim inf_{n} E^{s}(S^{g_{εn}} (v_{n}) ∪ Γ_{ε_{n},a_{n}}^{0}).

Since by assumption E^{s}(Γ_{ε_{n},a_{n}}^{0}) → E^{s}(Γ^{0}), we conclude that

E^{s}(S^{g}(v) \ Γ^{0}) ≤ lim inf_{n} E^{s}(S^{g_{εn}} (v_{n}) \ Γ_{ε_{n},a_{n}}^{0}).

We deduce that

E^{el}(0)(v) + E^{s}(S^{g}(v) \ Γ^{0}) ≤ lim inf_{n} [E^{el}(0)(v_{n}) + E^{s}(S^{g_{εn}} (v_{n}) \ Γ_{ε_{n},a_{n}}^{0})]

that is

F (v) ≤ lim inf_{n} F_{ε_{n},a_{n}}(v_{n}).

The Γ-liminf inequality holds, and so the proof is concluded. □

6. The discontinuous finite element approximation

In this section we construct a discrete approximation of the quasistatic evolution of brittle fractures proposed in [13] and described in the Preliminaries: the discretization is done both in space and time. Let us consider

g_{ε} ∈ W^{1,1}([0, T]; W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})), g_{ε}(t) ∈ AF_{ε}(Ω; ℝ^{2}) for all t ∈ [0, T],

where AF_{ε}(Ω; ℝ^{2}) is defined in (4.2). Let δ > 0, and let N_{δ} be the largest integer such that δ(N_{δ} − 1) < T; we set t_{i}^{δ} := iδ for 0 ≤ i ≤ N_{δ} − 1, t_{N_{δ}}^{δ} := T and g_{ε}^{δ,i} := g_{ε}(t_{i}^{δ}). Let Γ^{0} ∈ Γ_{ε,a}(Ω) be a preexisting crack in Ω, where Γ_{ε,a}(Ω) is defined in (5.2).

Proposition 6.1. Let ε > 0, a ∈]0, \frac{1}{2}[ and δ > 0 be fixed. Then for all i = 0, …, N_{δ} there exists u_{ε,a}^{δ,i} ∈ A_{ε,a}^{B}(Ω; ℝ^{2}) such that, setting

Γ_{ε,a}^{δ,i} := Γ^{0} ∪ ⋃_{r=0}^{i} S^{g_{ε}^{δ,r}}(u_{ε,a}^{δ,r}),

we have for all v ∈ A_{ε,a}^{B}(Ω; ℝ^{2})

(6.1)  E^{el}(0)(u_{ε,a}^{δ,0}) + E^{s} (S^{g_{ε}^{δ,0}}(u_{ε,a}^{δ,0}) \ Γ^{0}) ≤ E^{el}(0)(v) + E^{s} (S^{g_{ε}^{δ,0}}(v) \ Γ^{0}) ,

and for 1 ≤ i ≤ N_{δ}

(6.2)  E^{el}(t_{i}^{δ})(u_{ε,a}^{δ,i}) + E^{s} (S^{g_{ε}^{δ,i}}(u_{ε,a}^{δ,i}) \ Γ_{ε,a}^{δ,i−1}) ≤ E^{el}(t_{i}^{δ})(v) + E^{s} (S^{g_{ε}^{δ,i}}(v) \ Γ_{ε,a}^{δ,i−1}) .

Proof. Let u_{ε,a}^{δ,0} be a minimum of the following problem

(6.3)  min_{u∈A_{ε,a}^{B}(Ω;ℝ^{2})} {E^{el}(0)(u) + E^{s}(S^{g_{ε}^{δ,0}}(u) \ Γ^{0})} .

We set Γ_{ε,a}^{δ,0} := Γ^{0} ∪ S^{g_{ε}^{δ,0}}(u_{ε,a}^{δ,0}). Recursively, supposing to have constructed u_{ε,a}^{δ,i−1} and Γ_{ε,a}^{δ,i−1}, let u_{ε,a}^{δ,i} be a minimum for

(6.4)  min_{u∈A_{ε,a}^{B}(Ω;ℝ^{2})} {E^{el}(t_{i}^{δ})(u) + E^{s}(S^{g_{ε}^{δ,i}}(u) \ Γ_{ε,a}^{δ,i−1})} .

We set Γ_{ε,a}^{δ,i} := Γ_{ε,a}^{δ,i−1} ∪ S^{g_{ε}^{δ,i}}(u_{ε,a}^{δ,i}). It is clear by construction that (6.1) and (6.2) hold.

Let us prove that problem (6.4) admits a solution, problem (6.3) being similar. Let (u_{n})_{n∈ℕ} be a minimizing sequence for problem (6.4): since g_{ε}^{δ,i} is an admissible test function, we deduce that for n large

E^{el}(t_{i}^{δ})(u_{n}) + E^{s} (S^{g_{ε}^{δ,i}}(u_{n}) \ Γ_{ε,a}^{δ,i−1}) ≤ E^{el}(t_{i}^{δ})(g_{ε}^{δ,i}) + 1.

By the lower estimate on the elastic energy (3.20), we deduce that for n large

(6.5)  α_{0}^{E} (∥∇u_{n}∥_{p}^{p} + ∥u_{n}∥_{q}^{q}) + E^{s} (S^{g_{ε}^{δ,i}}(u_{n}) \ Γ_{ε,a}^{δ,i−1}) ≤ E^{el}(t_{i}^{δ})(g_{ε}^{δ,i}) + 1 + β_{0}^{E}.

Let us indicate by T_{n}^{1}, …, T_{n}^{k} the triangles of T(u_{n}). Up to a subsequence, there exists T = {T^{1}, …, T^{k}} ∈ T_{ε,a}(Ω) such that for all i = 1, …, k we have T_{n}^{i} → T^{i} in the Hausdorff metric (see Section 2 for a precise definition). Let us consider T^{i} ∈ T, and let T̃^{i} be contained in the interior of T^{i}. For n large enough, T̃^{i} is contained in the interior of T_{n}^{i}; moreover (u_{n})_{|T̃^{i}} is affine and by (6.5) we have ∫T̃^{i} |∇u_{n}|^{p} dx + ∥u_{n}∥_{L^{∞}(T̃^{i};ℝ^{2})} ≤ C, with C independent of n. We deduce that there exists a function u^{i} affine on T̃^{i} such that up to a subsequence u_{n} → u uniformly on T̃^{i}. Since T̃^{i} is arbitrary, it turns out that u^{i} is actually defined on T^{i} and

E^{el}(t_{i}^{δ})_{|T^{i}}(u^{i}) ≤ lim inf_{n} E^{el}(t_{i}^{δ})_{|T^{i}}(u_{n}^{i}),

where E^{el}(t_{i}^{δ})_{|T^{i}} denotes the restriction of E^{el}(t_{i}^{δ}) to the maps defined on T^{i}. Let u ∈ A_{ε,a}(Ω; ℝ^{2}) be such that u = u^{i} on T^{i} for every i = 1, …, k: we have

E^{el}(t_{i}^{δ})(u) ≤ lim inf_{n} E^{el}(t_{i}^{δ})(u_{n}).

On the other hand, it is easy to see that S^{g_{ε}^{δ,i}}(u) is contained in the Hausdorff limit of S^{g_{ε}^{δ,i}}(u_{n}), and so u ∈ A_{ε,a}^{B}(Ω; ℝ^{2}); moreover we deduce

E^{s} (S^{g_{ε}^{δ,i}}(u) \ Γ_{ε,a}^{δ,i−1}) ≤ lim inf_{n} E^{s} (S^{g_{ε}^{δ,i}}(u_{n}) \ Γ_{ε,a}^{δ,i−1}) .

We conclude that u is a minimum point for the problem (6.4), so that the proof is concluded. □

The following estimate on the total energy is essential in order to study the asymptotic behavior of the discrete evolution as δ → 0, ε → 0 and a → 0. Let us set u_{ε,a}^{δ}(t) := u_{ε,a}^{δ,i} for all t_{i}^{δ} ≤ t < t_{i+1}^{δ} and i = 0, …, N_{δ} − 1, u_{ε,a}^{δ}(T) = u_{ε,a}^{δ,N_{δ}}.

Proposition 6.2. For all 0 ≤ j ≤ i ≤ N_{δ} we have

(6.6) E(t_{i}^{δ})(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) ≤ E(t_{j}^{δ})(u_{ε,a}^{δ,j}, Γ_{ε,a}^{δ,j}) + ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂W(∇u_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ

− ∫_{t_{j}^{δ}}^{t_{i}^{δ}} Ḟ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂F(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

− ∫_{t_{j}^{δ}}^{t_{i}^{δ}} Ġ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂G(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ + e_{ε,a}^{δ},

where e_{ε,a}^{δ} → 0 as δ → 0 uniformly in ε and a.

Proof. By the minimality property (6.2), comparing u_{ε,a}^{δ,i} with u_{ε,a}^{δ,i−1} − g_{ε}^{δ,i−1} + g_{ε}^{δ,i} we get

(6.7) W(∇u_{ε,a}^{δ,i}) − F(t_{i}^{δ})(u_{ε,a}^{δ,i}) − G(t_{i}^{δ})(u_{ε,a}^{δ,i}) + E^{s}(S^{g_{i}^{δ}}(u_{ε,a}^{δ,i}) \ Γ_{i−1}^{δ})

≤ W(∇u_{ε,a}^{δ,i−1} − ∇g_{ε}^{δ,i−1} + ∇g_{ε}^{δ,i}) − F(t_{i}^{δ})(u_{ε,a}^{δ,i−1} − g_{ε}^{δ,i−1} + g_{ε}^{δ,i})
− G(t_{i}^{δ})(u_{ε,a}^{δ,i−1} − g_{ε}^{δ,i−1} + g_{ε}^{δ,i}).

We have

(6.8) W(∇u_{ε,a}^{δ,i−1} − ∇g_{ε}^{δ,i−1} + ∇g_{ε}^{δ,i}) = W(∇u_{ε,a}^{δ,i−1})

+ ⟨∂W(∇u_{ε,a}^{δ,i−1} + ϑ_{ε,a}^{δ,i−1}(∇g_{ε}^{δ,i} − ∇g_{ε}^{δ,i−1})), ∇g_{ε}^{δ,i} − ∇g_{ε}^{δ,i−1}⟩

= W(∇u_{ε,a}^{δ,i−1}) + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂W(∇u_{ε,a}^{δ}(τ) + v_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ,

where ϑ_{ε,a}^{δ,i−1} ∈]0, 1[ and v_{ε,a}^{δ}(τ) := ϑ_{ε,a}^{δ,i−1}(∇g_{ε}^{δ,i} − ∇g_{ε}^{δ,i−1}) for all τ ∈ [t_{i−1}^{δ}, t_{i}^{δ}[.

Similarly we obtain

(6.9) F(t_{i}^{δ})(u_{ε,a}^{δ,i−1} − g_{ε}^{δ,i−1} + g_{ε}^{δ,i}) = F(t_{i}^{δ})(u_{ε,a}^{δ,i−1}) + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂F(t_{i}^{δ})(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ,

and

(6.10) G(t_{i}^{δ})(u_{ε,a}^{δ,i−1} − g_{ε}^{δ,i−1} + g_{ε}^{δ,i}) = G(t_{i}^{δ})(u_{ε,a}^{δ,i−1}) + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂G(t_{i}^{δ})(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ,

where w_{ε,a}^{δ}(τ) := λ_{ε,a}^{δ,i−1}(g_{ε}^{δ,i} − g_{ε}^{δ,i−1}), z_{ε,a}^{δ}(τ) := ν_{ε,a}^{δ,i−1}(g_{ε}^{δ,i} − g_{ε}^{δ,i−1}) for all τ ∈ [t_{i−1}^{δ}, t_{i}^{δ}[, and λ_{ε,a}^{δ,i−1}, ν_{ε,a}^{δ,i−1} ∈]0, 1[.

Since by (3.10) we have for τ ∈ [t_{i−1}^{δ}, t_{i}^{δ}[

⟨∂F(t_{i}^{δ})(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩

= ∫_{τ}^{t_{i}^{δ}}⟨∂Ḟ(s)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ ds

we get by (3.13)

(6.11) |⟨∂F(t_{i}^{δ})(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩|
≤ ∫_{τ}^{t_{i}^{δ}} |⟨∂Ḟ(s)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| ds

≤ ∫_{τ}^{t_{i}^{δ}} [α_{4}^{F}(s)∥u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)∥q̇−1 ; q̇ + β_{4}^{F}(s)] ∥ġ_{ε}(τ)∥q̇ ds ≤ γ_{F}^{δ,ε,a}∥ġ_{ε}(τ)∥q̇,

where

γ_{F}^{δ,ε,a} := max_{1≤i≤N_{δ}} (∥u_{ε,a}^{δ,i−1} + λ_{ε,a}^{δ,i−1}(g_{ε}^{δ,i} − g_{ε}^{δ,i−1})∥q̇ ; q̇−1 ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} α_{4}^{F}(s) ds + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} β_{4}^{F}(s) ds) .

Similarly we obtain (6.12)

|⟨∂G(t_{i}^{δ})(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂G(τ)(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| ≤ γ_{G}^{δ,ε,a}∥ġ_{ε}(τ)∥_{r,∂_{S}Ω},

where

γ_{G}^{δ,ε,a} := max_{1≤i≤N_{δ}} (∥u_{ε,a}^{δ,i−1} + ν_{ε,a}^{δ,i−1}(g_{ε}^{δ,i} − g_{ε}^{δ,i−1})∥_{r,∂_{S}Ω}^{r−1} ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} a_{4}^{G}(s) ds + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} b_{4}^{G}(s) ds) .

From (6.7), taking into account (6.8), (6.9), (6.10), (6.11), (6.12), we have

(6.13) E(t_{i}^{δ})(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) ≤ E(t_{i−1}^{δ})(u_{ε,a}^{δ,i−1}, Γ_{ε,a}^{δ,i−1}) + ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂W(∇u_{ε,a}^{δ}(τ) + v_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ

− ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} Ḟ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

− ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} Ġ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}}⟨∂G(τ)(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

+ γ_{F}^{δ,ε,a} ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} ∥ġ_{ε}(τ)∥q̇ dτ + γ_{G}^{δ,ε,a} ∫_{t_{i−1}^{δ}}^{t_{i}^{δ}} ∥ġ_{ε}(τ)∥_{r,∂_{S}Ω} dτ.

Taking now 0 ≤ j ≤ i ≤ N_{δ}, summing in (6.13) form t_{j}^{δ} to t_{i}^{δ}, we obtain

(6.14) E(t_{i}^{δ})(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}) ≤ E(t_{j}^{δ})(u_{ε,a}^{δ,j}, Γ_{ε,a}^{δ,j}) + ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂W(∇u_{ε,a}^{δ}(τ) + v_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ

− ∫_{t_{j}^{δ}}^{t_{i}^{δ}} Ḟ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

− ∫_{t_{j}^{δ}}^{t_{i}^{δ}} Ġ(τ)(u_{ε,a}^{δ}(τ)) dτ − ∫_{t_{j}^{δ}}^{t_{i}^{δ}}⟨∂G(τ)(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

+ γ_{F}^{δ,ε,a} ∫_{t_{j}^{δ}}^{t_{i}^{δ}} ∥ġ_{ε}(τ)∥q̇ dτ + γ_{G}^{δ,ε,a} ∫_{t_{j}^{δ}}^{t_{i}^{δ}} ∥ġ_{ε}(τ)∥_{r,∂_{S}Ω} dτ.

Setting

(6.15) e_{ε,a}^{δ} := ∫_{0}^{1} |⟨∂W(∇u_{ε,a}^{δ}(τ) + v_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ − ⟨∂W(∇u_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩| dτ

+ ∫_{0}^{1} |⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂F(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| dτ
+ ∫_{0}^{1} |⟨∂G(τ)(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂G(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| dτ

+ γ_{F}^{δ,ε,a} ∫_{0}^{1} ∥ġ_{ε}(τ)∥q̇ dτ + γ_{G}^{δ,ε,a} ∫_{0}^{1} ∥ġ_{ε}(τ)∥_{r,∂_{S}Ω} dτ,

from (6.14) we formally obtain (6.6). Let us prove that e_{|eps,a}^{δ} → 0 as δ → 0 uniformly in ε and a. By (6.2), comparing u_{ε,a}^{δ,i} with g_{ε}^{δ,i}, and taking into account (3.20), we get for all i = 1, …, N_{δ},

∥∇u_{ε,a}^{δ,i}∥_{p} + ∥u_{ε,a}^{δ,i}∥_{q} ≤ C′,

where

C′ := \frac{1}{α_{0}^{E}} max_{i=0,...,N_{δ}} (E^{el}(t_{i}^{δ})(g_{ε}^{δ,i}) + β_{0}^{E}) .

Since Ω_{S} is Lipschitz, there exists K_{S} > 0 depending only on p, q such that

∥u∥_{p,Ω_{S}} ≤ K_{S}(∥∇u∥_{p,Ω_{S}} + ∥u∥_{q,Ω_{S}})

for all u ∈ W^{1,p}(Ω_{S}; ℝ^{2}) ∩ L^{q}(Ω_{S}; ℝ^{2}). Taking into account (3.15), we obtain

∥u_{ε,a}^{δ,i}∥_{r,∂_{S}Ω} ≤ C′′

for some C′′ independent of δ. Since g_{ε} ∈ W^{1,1}([0, T]; W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})), we obtain that for all τ ∈ [0, T] as δ → 0

v_{ε,a}^{δ}(τ) → 0 strongly in L^{p}(Ω; M^{2×2}),
w_{ε,a}^{δ}(τ) → 0 strongly in L^{q}(Ω; ℝ^{2}),
z_{ε,a}^{δ}(τ) → 0 strongly in L^{r}(∂_{S}Ω; ℝ^{2}).

Moreover γ_{F}^{δ,ε,a} → 0 and γ_{G}^{δ,ε,a} → 0 as δ → 0. Finally, by [13, Lemma 4.9], as δ → 0 we have that for all τ ∈ [0, T]

|⟨∂W(∇u_{ε,a}^{δ}(τ) + v_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ − ⟨∂W(∇u_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩| → 0,
|⟨∂F(τ)(u_{ε,a}^{δ}(τ) + w_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂F(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| → 0,
|⟨∂G(τ)(u_{ε,a}^{δ}(τ) + z_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ − ⟨∂G(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩| → 0,

uniformly in ε, a. By the Dominated Convergence Theorem, we conclude that e_{ε,a}^{δ} → 0 as δ → 0 uniformly in ε and a, and the proof is finished. □

7. The approximation result

In this section we study the asymptotic behavior of the discrete evolution obtained in Section 6. Let us consider a given initial crack Γ^{0} ∈ Γ(Ω) where Γ(Ω) is defined as in (5.1), and a boundary deformation g ∈ W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})). Let Γ_{ε,a}^{0} ∈ Γ_{ε,a}(Ω) be an approximation of Γ^{0} in the sense of Proposition 5.1, and let us consider

g_{ε} ∈ W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})),

such that

g_{ε}(t) ∈ AF_{ε}(Ω; ℝ^{2}) for all t ∈ [0, T],

and such that for ε → 0

g_{ε} → g strongly in W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})).

Let

{(u_{ε,a}^{δ,i}, Γ_{ε,a}^{δ,i}), i = 0, …, N_{δ}}

be the discrete evolution relative to the initial crack Γ_{ε,a}^{0} and boundary data g_{ε} given by Proposition 6.1. We make the following piecewise constant interpolation in time:

(7.1)  u_{ε,a}^{δ}(t) := u_{ε,a}^{δ,i},  Γ_{ε,a}^{δ}(t) := Γ_{ε,a}^{δ,i},  g_{ε}^{δ}(t) := g_{ε}(t_{i}^{δ})  for t_{i}^{δ} ≤ t < t_{i+1}^{δ},

i = 0, …, N_{δ} − 1, and u_{ε,a}^{δ}(T) := u_{ε,a}^{δ,N_{δ}}, Γ_{ε,a}^{δ}(T) := Γ_{ε,a}^{δ,N_{δ}}, g_{ε}^{δ}(T) := g_{ε}(T).

By Proposition 6.2, for all v ∈ A_{ε,a}^{B}(Ω; ℝ^{2}) we have

E^{el}(0)(u_{ε,a}^{δ}(0)) + E^{s} (S^{g_{ε}^{δ}(0)}(u_{ε,a}^{δ}(0)) \ Γ_{ε,a}^{0}) ≤ E^{el}(0)(v) + E^{s} (S^{g_{ε}^{δ}(0)}(v) \ Γ_{ε,a}^{0}) ,

and for all t ∈ [t_{i}^{δ}, t_{i+1}^{δ}[ and for all v ∈ A_{ε,a}^{B}(Ω; ℝ^{2})

(7.2)  E^{el}(t_{i}^{δ})(u_{ε,a}^{δ}(t)) ≤ E^{el}(t_{i}^{δ})(v) + E^{s} (S^{g_{ε}^{δ}(t)}(v) \ Γ_{ε,a}^{δ}(t)) .

Here E^{el} and E^{s} are defined in (3.19) and (3.1) respectively. Finally for all 0 ≤ s ≤ t ≤ 1 we have

(7.3)  E(t_{i}^{δ})(u_{ε,a}^{δ}(t), Γ_{ε,a}^{δ}(t)) ≤E(s_{j}^{δ})(u_{ε,a}^{δ}(s), Γ_{ε,a}^{δ}(s)) + ∫_{s_{j}^{δ}}^{t_{i}^{δ}}⟨∂W(∇u_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ

− ∫_{s_{j}^{δ}}^{t_{i}^{δ}} Ḟ(τ)(u_{ε,a}^{δ}(τ)) − ∫_{s_{j}^{δ}}^{t_{i}^{δ}}⟨∂F(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ

− ∫_{s_{j}^{δ}}^{t_{i}^{δ}} Ġ(τ)(u_{ε,a}^{δ}(τ)) − ∫_{s_{j}^{δ}}^{t_{i}^{δ}}⟨∂G(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ + e_{ε,a}^{δ},

where s_{j}^{δ} ≤ s < s_{j+1}^{δ} and t_{i}^{δ} ≤ t < t_{i+1}^{δ}, e_{ε,a}^{δ} is defined as in (6.15), and E(t)(u, Γ) is as in (3.18). Recall that e_{ε,a}^{δ} → 0 as δ → 0 uniformly in ε, a.

Comparing u_{ε,a}^{δ}(t) with g_{ε}^{δ}(t) by (7.2), and in view of (3.7), (3.11), (3.12), (3.16), (3.17), (6.1) and (3.2), by (7.3) with s = 0 we deduce that there exists C′ ∈]0, +∞[ such that for all t, δ, ε and a

(7.4)  ∥∇u_{ε,a}^{δ}(t)∥_{p} + ∥u_{ε,a}^{δ}(t)∥_{q} + H^{1}(Γ_{ε,a}^{δ}(t)) ≤ C′.

By the time dependence of E^{el}(·, ·), in view of (7.4), by (7.2) and (7.3) we have that there exists o_{ε,a}^{δ} → 0 as δ, ε → 0 uniformly in a such that for all t ∈ [0, T] and for all v ∈ A_{ε,a}^{B}(Ω; ℝ^{2})

(7.5)  E^{el}(t)(u_{ε,a}^{δ}(t)) ≤ E^{el}(t)(v) + E^{s} (S^{g_{ε}^{δ}(t)}(v) \ Γ_{ε,a}^{δ}(t)) + o_{ε,a}^{δ},

and for all 0 ≤ s ≤ t ≤ T

(7.6)  E(t)(u_{ε,a}^{δ}(t), Γ_{ε,a}^{δ}(t)) ≤E(s)(u_{ε,a}^{δ}(s), Γ_{ε,a}^{δ}(s)) + ∫_{s}^{t}⟨∂W(∇u_{ε,a}^{δ}(τ)), ∇ġ_{ε}(τ)⟩ dτ
− ∫_{s}^{t} Ḟ(τ)(u_{ε,a}^{δ}(τ)) − ∫_{s}^{t}⟨∂F(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ
− ∫_{s}^{t} Ġ(τ)(u_{ε,a}^{δ}(τ)) − ∫_{s}^{t}⟨∂G(τ)(u_{ε,a}^{δ}(τ)), ġ_{ε}(τ)⟩ dτ + o_{ε,a}^{δ}.

Inequality (7.4) gives a natural precompactness of (u_{ε,a}^{δ}(t)) in GSBV_{q}^{p}(Ω; ℝ^{2}). The main result of the paper is the following.

Theorem 7.1. Let δ > 0, ε > 0, a ∈]0, \frac{1}{2}[, and let \{t → (u_{ε,a}^{δ}(t), Γ_{ε,a}^{δ}(t)) : t ∈ [0, T]\} be the discrete evolution given by (7.1) relative to the initial crack Γ_{ε,a}^{0} and the boundary data g_{ε}. Then there exist a quasistatic evolution {t → (u(t), Γ(t))} in the sense of Theorem 3.1 and sequences δ_{n} → 0, ε_{n} → 0, a_{n} → 0, such that setting u_{n}(t) := u_{ε_{n},a_{n}}^{δ_{n}}(t) and Γ_{n}(t) := Γ_{ε_{n},a_{n}}^{δ_{n}}(t), for all t ∈ [0, T] the following facts hold.

(a) For every t ∈ [0, T], (u_{n}(t))_{n∈ℕ} is weakly precompact in GSBV_{q}^{p}(Ω; ℝ^{2}), and every accumulation point ũ(t) is such that S^{g(t)}(ũ(t)) ˜⊂ Γ(t),

(7.7)  E^{el}(t)(ũ(t)) ≤ E^{el}(t)(v) + E^{s} (S^{g(t)}(v) \ Γ(t))

for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, and

E^{el}(t)(u_{n}(t)) → E^{el}(t)(ũ(t)).

Moreover there exists a subsequence of (δ_{n}, ε_{n}, a_{n})_{n∈ℕ} (depending on t) such that

u_{n}(t) ⇀ u(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

(b) For every t ∈ [0, T] we have

(7.8)  E(t)(u_{n}(t), Γ_{n}(t)) → E(t)(u(t), Γ(t));

more precisely elastic and surface energies converge separately, that is

(7.9)  E^{el}(t)(u_{n}(t)) → E^{el}(t)(u(t)),  E^{s}(Γ_{n}(t)) → E^{s}(Γ(t)).

For the proof of Theorem 7.1 we need two preliminary steps. First of all, we fix a and study the asymptotic for δ, ε → 0 (Lemma 7.2), and then we let a → 0 using a diagonal argument (Lemma 7.4).

Lemma 7.2. Let a be fixed, t ∈ [0, T], and let δ_{n} → 0 and ε_{n} → 0. There exists Γ_{a}(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and a subsequence of (δ_{n}, ε_{n})_{n∈ℕ} (which we denote with the same symbol), such that the following facts hold:

(a) if w_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) is such that S^{g_{εn}^{δn} (t)}(w_{n}) ⊆ Γ_{ε_{n},a}^{δ_{n}}(t) and
w_{n} ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}),

then we have

S^{g(t)}(w) ˜⊂ Γ_{a}(t);

(b) there exists μ(a) with μ(a) → 1 as a → 0 such that for every accumulation point u_{a}(t) of (u_{ε_{n},a}^{δ_{n}}(t))_{n∈ℕ} for the weak convergence in GSBV_{q}^{p}(Ω; ℝ^{2}) and for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, we have

(7.10)  E^{el}(t)(u_{a}(t)) ≤ E^{el}(t)(v) + μ(a)E^{s} (S^{g(t)}(v) \ Γ_{a}(t));

moreover

(7.11)  lim_{n} E^{el}(t)(u_{ε_{n},a}^{δ_{n}}(t)) = E^{el}(t)(u_{a}(t));

(c) we have

E^{s}(Γ_{a}(t)) ≤ lim inf_{n} E^{s}(Γ_{ε_{n},a}^{δ_{n}}(t)).

Proof. We now perform a variant of [13, Theorem 4.7]. Let (φ_{k})_{k∈ℕ} ⊆ L^{1}(Ω; ℝ^{2}) be dense in L^{1}(Ω; ℝ^{2}). For every φ_{k} and for every m ∈ ℕ, let v_{k,m}^{n,a}(t) be a minimum of the problem

min{∥∇v∥_{p} + ∥v∥_{q} + m∥v − φ_{k}∥_{1} : v ∈ V_{a}^{n}},

where

V_{a}^{n} := {v ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}), S^{g_{εn}^{δn} (t)}(v) ⊆ Γ_{ε_{n},a}^{δ_{n}}(t)}.

Since by (7.4) we have H^{1}(Γ_{ε_{n},a}^{δ_{n}}(t)) ≤ C′, by Theorem 2.1 there exists a subsequence of (δ_{n}, ε_{n})_{n∈ℕ} (which we denote with the same symbol) such that v_{k,m}^{n,a}(t) weakly converges to some v_{k,m}^{a}(t) ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) as n → +∞ for all k, m ∈ ℕ. We set

(7.12)  Γ_{a}(t) := ⋃_{k,m} S^{g(t)}(v_{k,m}^{a}(t)).

Let us see that Γ_{a}(t) satisfies all the properties of the lemma. Clearly Γ_{a}(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and point (c) is a consequence of Theorem 2.1. In particular by (7.4) we have that

(7.13)  H^{1}(Γ_{a}(t)) ≤ C′.

Let us come to point (a). Let w_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) be such that S^{g_{εn}^{δn} (t)}(w_{n}) ⊆ Γ_{ε_{n},a}^{δ_{n}}(t) and

w_{n} ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

We claim that there exists k_{m} → +∞ such that

(7.14)  v_{k_{m},m}^{a}(t) ⇀ w  weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

Then since S^{g(t)}(v_{k_{m},m}^{a}(t)) ⊆ Γ_{a}(t) for all m and in view of (7.13), we deduce that S^{g(t)}(w) ˜⊂ Γ_{a}(t). Let us prove (7.14). Fixed m ∈ ℕ, let us choose k_{m} in such a way that

m∥w − φ_{k_{m}}∥_{1} → 0.

By minimality of v_{k_{m},m}^{n,a}(t) we have for all n

∥∇v_{k_{m},m}^{n,a}(t)∥_{p} + ∥v_{k_{m},m}^{n,a}(t)∥_{q} + m∥v_{k_{m},m}^{n,a}(t) − φ_{k_{m}}∥_{1} ≤ ∥∇w_{n}∥_{p} + ∥w_{n}∥_{q} + m∥w_{n} − φ_{k_{m}}∥_{1}.

Passing to the limit in n, by lower semicontinuity we get for some C ≥ 0

∥∇v_{k_{m},m}^{a}(t)∥_{p} + ∥v_{k_{m},m}^{a}(t)∥_{q} + m∥v_{k_{m},m}^{a}(t) − φ_{k_{m}}∥_{1} ≤ C + m∥w − φ_{k_{m}}∥_{1}.

We deduce for m → +∞

∥v_{k_{m},m}^{a}(t) − φ_{k_{m}}∥_{1} → 0,

which together with ∥φ_{k_{m}} − w∥_{1} → 0 implies that

v_{k_{m},m}^{a}(t) → w strongly in L^{1}(Ω; ℝ^{2}).

Since

∥∇v_{k_{m},m}^{a}(t)∥_{p} + ∥v_{k_{m},m}^{a}(t)∥_{q} ≤ C + m∥w − φ_{k_{m}}∥_{1} ≤ C + 1

for m large, we have that v_{k_{m},m}^{a}(t) ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}), and this proves (7.14).

Finally, let us come to point (b). Let v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ⊆ \overline{Ω}_{B}, and let us fix k_{1}, …, k_{s} and m_{1}, …, m_{r} in ℕ. By Proposition 4.2, there exists v_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) such that

lim_{n} E^{el}(t)(v_{n}) = E^{el}(t)(v)

and

lim sup_{n} E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t)) ≤ lim sup_{n} E^{s}(S^{g_{εn}^{δn} (t)}(v_{n}) \ ⋃_{i≤s, j≤r} S(v_{k_{i},m_{j}}^{n,a}))

≤ μ(a)E^{s}(S^{g(t)}(v) \ ⋃_{i≤s, j≤r} S(v_{k_{i},m_{j}}^{a})),

where μ(a) → 1 as a → 0. Since the k_{i}’s and the m_{j}’s are arbitrary, we obtain that

(7.15)  lim sup_{n} E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t)) ≤ μ(a)E^{s} (S^{g(t)}(v) \ Γ_{a}(t)) .

Let us suppose that u_{ε_{n},a}^{δ_{n}}(t) ⇀ u_{a}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}) along a suitable subsequence which we indicate by the same symbol. By the minimality property (7.5), comparing u_{ε_{n},a}^{δ_{n}}(t) with v_{n} we get

(7.16)  E^{el}(t)(u_{ε_{n},a}^{δ_{n}}(t)) ≤ E^{el}(t)(v_{n}) + E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t)) + o_{n},

with o_{n} → 0 as n → +∞. Then we have

E^{el}(t)(u_{a}(t)) ≤ lim inf_{n} E^{el}(t)(u_{ε_{n},a}^{δ_{n}}(t))

≤ lim sup_{n} (E^{el}(t)(v_{n}) + E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t)))

≤ E^{el}(t)(v) + lim sup_{n} E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t))

≤ E^{el}(t)(v) + μ(a)E^{s} (S^{g(t)}(v) \ Γ_{a}(t)) ,

that is (7.10) holds. Choosing v = u_{a}(t), passing to the limsup in (7.16), and taking into account (7.15) we obtain that

lim sup_{n} E^{el}(t)(u_{ε_{n},a}^{δ_{n}}(t)) ≤ E^{el}(t)(u_{a}(t)).

Since by (7.10) E^{el}(t)(u_{a}(t)) is independent of the accumulation point u_{a}(t), we conclude that (7.11) holds. □

Remark 7.3. Using Lemma 7.2, it is possible to construct an increasing family {t → Γ_{a}(t) : t ∈ [0, T]} and a subsequence of (δ_{n}, ε_{n})_{n∈ℕ} such that points (a), (b) and (c) of Lemma 7.2 hold for every t ∈ [0, T]. This evolution {t → Γ_{a}(t) : t ∈ [0, T]} can be considered as an approximate quasistatic evolution, in the sense that it satisfies irreversibility, but satisfies static equilibrium and nondissipativity up to a small error due to the fact that a is kept fixed. The presence of μ(a) in the minimality property (7.10) takes into account the anisotropy in the approximation of the surface energy: in fact, since a is kept fixed, the adaptive edges of the triangulations T_{ε,a}(Ω) cannot recover all the possible directions. The nondissipativity condition up to a small error can be obtained using the minimality property (7.10) and following [13, Theorem 3.13] (estimate from below of the total energy).

The construction of {t → Γ_{a}(t) : t ∈ [0, T]} is the following. If D ⊆ [0, T] is countable and dense, by Lemma 7.2 and using a diagonalization argument, we can find a subsequence of (δ_{n}, ε_{n})_{n∈ℕ} and an increasing family Γ_{a}(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω), t ∈ D, such that points (a), (b) and (c) hold for every t ∈ D. Let us set for every t ∈ [0, T]

Γ_{a}^{+}(t) := ⋂_{s≥t,s∈D} Γ_{a}(s).

Clearly {t → Γ_{a}^{+}(t) : t ∈ [0, T]} is increasing, in the sense that Γ_{a}(s) ˜⊂ Γ_{a}(t) for all s ≤ t. As a consequence, the set J of discontinuity points of H^{1}(Γ_{a}^{+}(t)) is at most countable. We can extract a further subsequence of (δ_{n}, ε_{n})_{n∈ℕ} such that Γ_{a}(t) is determined also for all t ∈ J (notice that Γ_{a}(t) ˜⊂ Γ_{a}^{+}(t)). For all t ∈ [0, T]\(D∪J) we set Γ_{a}(t) := Γ_{a}^{+}(t). We have that Γ_{a}(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and {t → Γ_{a}(t) : t ∈ [0, T]} is increasing.

For t ∈ D ∪ J, Γ_{a}(t) satisfies by construction points (a), (b) and (c) of Lemma 7.2. Let us consider the case t ∈ [0, T] \ (D ∪ J).

Concerning point (a), we have that S^{g(t)}(w) ˜⊂ Γ_{a}(s) for every s ∈ D ∩ [t, T], so that passing to the intersection we get S^{g(t)}(u_{a}(t)) ˜⊂ Γ_{a}(t).

As for point (b), considering s ∈ D ∩ [0, t[, for every v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, we have there exists v_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) such that

lim_{n} E^{el}(t)(v_{n}) = E^{el}(t)(v),

and

lim sup_{n} E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(t)) ≤ lim sup_{n} E^{s} (S^{g_{εn}^{δn} (t)}(v_{n}) \ Γ_{ε_{n},a}^{δ_{n}}(s))

≤ μ(a)E^{s} (S^{g(t)}(v) \ Γ_{a}(s)) .

Then by minimality property (7.5) and passing to the limit in n we have

E^{el}(t)(u(t)) ≤ E^{el}(t)(v) + μ(a)E^{s} (S^{g(t)}(v) \ Γ_{a}(s)) .

Letting s → t we get that (7.10) holds. Reasoning as in Lemma 7.2, we get that also (7.11) holds. Finally, coming to point (c), we have that for all s ∈ D ∩ [0, t[

lim inf_{n} E^{s}(Γ_{ε_{n},a}^{δ_{n}}(t)) ≥ lim inf_{n} E^{s}(Γ_{ε_{n},a}^{δ_{n}}(s)) ≥ E^{s}(Γ_{a}(s)),

so that letting s ↗ t, and recalling that t is a continuity point for E^{s}(Γ_{ε_{n},a}^{δ_{n}}(·)), we obtain that the lower semicontinuity holds.

We can now let a → 0.

Lemma 7.4. There exist a map \{t → Γ(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω), t ∈ [0, T]\} and sequences δ_{n} → 0, ε_{n} → 0, a_{n} → 0 such that the following facts hold:

(a) Γ^{0} ˜⊂ Γ(s) ˜⊂ Γ(t) for all 0 ≤ s ≤ t ≤ T;
(b) for all t ∈ [0, T], if w_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) with S^{g_{εn}^{δn} (t)}(w_{n}) ⊆ Γ_{ε_{n},a_{n}}^{δ_{n}}(t) is such that
w_{n} ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}),

then we have

S^{g(t)}(w) ˜⊂ Γ(t);

(c) for all t ∈ [0, T] and for every accumulation point u(t) of (u_{ε_{n},a_{n}}^{δ_{n}}(t))_{n∈ℕ} for the weak convergence in GSBV_{q}^{p}(Ω; ℝ^{2}) and for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, we have

(7.17)  E^{el}(t)(u(t)) ≤ E^{el}(t)(v) + E^{s} (S^{g(t)}(v) \ Γ(t)) ,

and

(7.18)  E^{el}(t)(u(t)) = lim_{n} E^{el}(t)(u_{ε_{n},a_{n}}^{δ_{n}}(t));

(d) for all t ∈ [0, T] we have

(7.19)  E^{s}(Γ(t)) ≤ lim inf_{n} E^{s}(Γ_{ε_{n},a_{n}}^{δ_{n}}(t)).

Proof. Let us consider δ_{h} → 0 and ε_{h} → 0. Given a ∈]0, \frac{1}{2}[ and t ∈ [0, T], let Γ_{a}(t) be the rectifiable set given by Lemma 7.2. Recall that by (7.12) we have

Γ_{a}(t) = ⋃_{k,m} S^{g(t)}(v_{k,m}^{a}(t)),

where v_{k,m}^{a}(t) is the weak limit in GSBV_{q}^{p}(Ω; ℝ^{2}) along a suitable subsequence depending on a of a minimum v_{k,m}^{h,a}(t) of the problem

(7.20)  min{∥∇v∥_{p} + ∥v∥_{q} + m∥v − φ_{k}∥_{1} : v ∈ V_{a}^{h}(t)},

where (φ_{k})_{k∈ℕ} ⊆ L^{1}(Ω; ℝ^{2}) is dense in L^{1}(Ω; ℝ^{2}) and

V_{a}^{h}(t) := {v ∈ A_{ε,a}^{B}(Ω′; ℝ^{2}), S^{g_{ε_{h}}^{δ_{h}} (t)}(v) ⊆ Γ_{ε_{h},a}^{δ_{h}}(t)}.

Let a_{n} → 0, and let D := {t_{j} : j ∈ ℕ} ⊆ [0, T] be countable and dense with 0 ∈ D. Using a diagonal argument, up to a subsequence of (δ_{h}, ε_{h})_{h∈ℕ}, we may suppose that for all t ∈ D and for all n

v_{k,m}^{h,a_{n}}(t) ⇀ v_{k,m}^{a_{n}}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

Moreover, we may assume that for all t ∈ D and for all n

u_{ε_{h},a_{n}}^{δ_{h}}(t) ⇀ u_{a_{n}}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2})

with

E^{el}(t)(u_{ε_{h},a_{n}}^{δ_{h}}(t)) → E^{el}(t)(u_{a_{n}}(t)).

By Lemma 7.2, we have that u_{a_{n}}(t) satisfies the minimality property (7.10).

Up to a subsequence of (a_{n})_{n∈ℕ}, we may suppose that for all k, m and t ∈ D we have

(7.21)  v_{k,m}^{a_{n}}(t) ⇀ v_{k,m}(t)  weakly in GSBV_{q}^{p}(Ω; ℝ^{2}),

and

(7.22)  u_{a_{n}}(t) ⇀ u(t)  weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

For all t ∈ D, let us set

(7.23)  Γ(t) := ⋃_{k,m} S^{g(t)}(v_{k,m}(t)).

By Proposition 4.2, in view of the minimality property (7.10) and taking into account that μ(a_{n}) → 1, we have that for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}

(7.24)  E^{el}(t)(u(t)) ≤ E^{el}(t)(v) + E^{s}(S^{g(t)}(v) \ Γ(t)),

and as a consequence, we obtain

E^{el}(t)(u_{a_{n}}(t)) → E^{el}(t)(u(t)).

We now perform the following diagonal argument. Choose δ_{h_{0}}, ε_{h_{0}} in such a way that

∥v_{0,0}^{h_{0},a_{0}}(t_{0}) − v_{0,0}^{a_{0}}(t_{0})∥_{1} + ∥u_{ε_{h_{0}} ,a_{0}}^{δ_{h_{0}}}(t_{0}) − u_{a_{0}}(t_{0})∥_{1}

+ |E^{el}(t_{0})(u_{ε_{h_{0}} ,a_{0}}^{δ_{h_{0}}}(t_{0})) − E^{el}(t_{0})(u_{a_{0}}(t_{0}))| ≤ 1.

Supposing to have constructed δ_{h_{n}}, ε_{h_{n}}, we choose δ_{h_{n+1}}, ε_{h_{n+1}} in such a way that for all k ≤ n+1, m ≤ n + 1 and for all t_{i} with 1 ≤ i ≤ n + 1 we have

∥v_{k,m}^{h_{n+1},a_{n+1}}(t_{i}) − v_{k,m}^{a_{n+1}}(t_{i})∥_{1} + ∥u_{ε_{h_{n+1}} ,a_{n+1}}^{δ_{h_{n+1}}}(t_{i}) − u_{a_{n+1}}(t_{i})∥_{1}

+ |E^{el}(t_{i})(u_{ε_{h_{n+1}} ,a_{n+1}}^{δ_{h_{n+1}}}(t_{i})) − E^{el}(t_{i})(u_{a_{n+1}}(t_{i}))| ≤ \frac{1}{n + 1}.

Let us set δ_{n} := δ_{h_{n}} and ε_{n} := ε_{h_{n}}, and let us prove that Γ(t) defined in (7.23) satisfies the properties of the Lemma. We have immediately that Γ(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω).

Concerning point (d), notice that

Γ_{ε_{n},a_{n}}^{δ_{n}}(t) = ⋃_{m,k} S^{g_{εn}^{δn} (t)}(v_{k,m}^{h_{n},a_{n}}(t)), Γ(t) = ⋃_{m,k} S^{g(t)}(v_{k,m}(t)),

and that for all k, m

v_{k,m}^{h_{n},a_{n}}(t) ⇀ v_{k,m}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2});

then (7.19) is a consequence of Theorem 2.1. In particular, by (7.4), we get that

(7.25)  H^{1}(Γ(t)) ≤ C′.

Let us come to point (b). Let w_{n} ∈ A_{ε_{n},a}^{B}(Ω; ℝ^{2}) with S^{g_{εn}^{δn} (t)}(w_{n}) ⊆ Γ_{ε_{n},a_{n}}^{δ_{n}}(t) be such that

w_{n} ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

For every m ∈ ℕ, let us choose k_{m} in such a way that

m∥w − φ_{k_{m}}∥_{1} → 0.

By minimality of v_{k_{m},m}^{h_{n},a_{n}}(t) we have for all n

∥∇v_{k_{m},m}^{h_{n},a_{n}}(t)∥_{p} + ∥v_{k_{m},m}^{h_{n},a_{n}}(t)∥_{q} + m∥v_{k_{m},m}^{h_{n},a_{n}}(t) − φ_{k_{m}}∥_{1} ≤ ∥∇w_{n}∥_{p} + ∥w_{n}∥_{q} + m∥w_{n} − φ_{k_{m}}∥_{1}.

By construction of h_{n}, and in view of (7.21), we have

v_{k_{m},m}^{h_{n},a_{n}}(t) ⇀ v_{k_{m},m}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

Then passing to the limit in n, by lower semicontinuity we get for some C ≥ 0

∥∇v_{k_{m},m}(t)∥_{p} + ∥v_{k_{m},m}(t)∥_{q} + m∥v_{k_{m},m}(t) − φ_{k_{m}}∥_{1} ≤ C + m∥w − φ_{k_{m}}∥_{1}.

We deduce for m → +∞

∥v_{k_{m},m}(t) − φ_{k_{m}}∥_{1} → 0,

which together with ∥φ_{k_{m}} − w∥_{1} → 0 implies that

v_{k_{m},m}(t) → w strongly in L^{1}(Ω; ℝ^{2}).

Since

∥∇v_{k_{m},m}(t)∥_{p} + ∥v_{k_{m},m}(t)∥_{q} ≤ C + m∥w − φ_{k_{m}}∥_{1} ≤ C + 1

for m large, we have that

v_{k_{m},m}(t) ⇀ w weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

Since S^{g(t)}(v_{k_{m},m}(t)) ⊆ Γ(t) for all m, and since H^{1}(Γ(t)) < C′, we deduce that S^{g(t)}(w) ˜⊂ Γ(t).

Coming to point (c), we have that (7.18) holds by construction. Moreover (7.17) holds in view of (7.24) and by the fact that u_{ε_{n},a_{n}}^{δ_{n}}(t) weakly converges in GSBV_{q}^{p}(Ω; ℝ^{2}) to u(t) defined in (7.22).

In order to prove point (a), notice that if s ≤ t with s, t ∈ D, we have for all k, m ∈ ℕ that

S^{g_{εn}^{δn} (t)}(v_{k,m}^{h_{n},a_{n}}(s) + g_{ε_{n}}^{δ_{n}}(t) − g_{ε_{n}}^{δ_{n}}(s)) ⊆ Γ_{ε_{n},a_{n}}^{δ_{n}}(s) ⊆ Γ_{ε_{n},a_{n}}^{δ_{n}}(t),

and

v_{k,m}^{h_{n},a_{n}}(s) + g_{ε_{n}}^{δ_{n}}(t) − g_{ε_{n}}^{δ_{n}}(s) ⇀ v_{k,m}(s) + g(t) − g(s) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}),

where v_{k,m}^{h,a}(s) and v_{k,m}(s) are defined in (7.20) and (7.21). By point (b) we deduce that

S^{g(t)}(v_{k,m}(s) + g(t) − g(s)) = S^{g(s)}(v_{k,m}(s)) ˜⊂ Γ(t).

Then by the definition of Γ(s) we get Γ(s) ˜⊂ Γ(t). Finally, by the same argument, we deduce Γ^{0} ˜⊂ Γ(s).

In order to deal with all t ∈ [0, T], we proceed as in Remark 7.3. For all t ∈ [0, T] \ D let us set

Γ^{+}(t) := ⋂_{s≥t,s∈D} Γ(s).

Clearly Γ^{+}(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and satisfies point (a), so that the set J of discontinuity points of H^{1}(Γ^{+}(·)) is at most countable. We can then extract a further subsequence of (δ_{n}, ε_{n}, a_{n})_{n∈ℕ} such that Γ(t) is determined also for all t ∈ J \ D (notice that Γ(t) ˜⊂ Γ^{+}(t)). For all t ∈ [0, T] \ (D ∪ J) we set Γ(t) := Γ^{+}(t). We have that Γ(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω) and that Γ(t) satisfies point (a). Let us see that Γ(t) satisfies also points (b), (c) and (d) also for t ∈ [0, T] \ (D ∪ J).

Concerning point (b), for every accumulation point u(t) of (u_{ε_{n},a_{n}}^{δ_{n}}(t))_{n∈ℕ} for the weak convergence in GSBV_{q}^{p}(Ω; ℝ^{2}), by the first part of the proof, we have that S^{g(t)}(u(t)) ˜⊂ Γ(s) for all s ∈ D with s ≥ t, so that passing to the intersection, we get that S^{g(t)}(u(t)) ˜⊂ Γ(t).

Let us come to point (c). Let

u_{j}(t) := u_{ε_{n_{j}} ,a_{n_{j}}}^{δ_{n_{j}}} (t) ⇀ u(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2})

along a subsequence n_{j} ↗ +∞. Let us set Γ_{j} := Γ_{ε_{n_{j}} ,a_{n_{j}}}^{δ_{n_{j}}} and g_{j} := g_{ε_{n_{j}}}^{δ_{n_{j}}}. Up to a further

subsequence there exists s_{j} ∈ D with s_{j} ↗ t, and such that setting u_{j}(s_{j}) := u_{ε_{n_{j}} ,a_{n_{j}}}^{δ_{n_{j}}} (s_{j}), we have

(7.26)  ∥u_{j}(s_{j}) − u(s_{j})∥_{1} + |E^{el}(s_{j})(u_{j}(s_{j})) − E^{el}(s_{j})(u(s_{j}))| → 0.

We have that there exists u^{∗}(t) ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) such that up to a subsequence

u(s_{j}) ⇀ u^{∗}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}).

By the minimality property (7.17) of u(s_{j}), for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}, we have that

E^{el}(s_{j})(u(s_{j})) ≤ E^{el}(s_{j})(v − g(t) + g(s_{j})) + E^{s}(S^{g(t)}(v) \ Γ(s_{j})).

Passing to the limit in j we have that for all v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) with S(v) ˜⊂ \overline{Ω}_{B}

(7.27)  E^{el}(t)(u^{∗}(t)) ≤ E^{el}(t)(v) + E^{s}(S^{g(t)}(v) \ Γ(t)).

As a consequence of the stability of this unilateral minimality property, it follows that

E^{el}(s_{j})(u(s_{j})) → E^{el}(t)(u^{∗}(t)).

By (7.26) we get

u_{j}(s_{j}) ⇀ u^{∗}(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}),

and

(7.28)  E^{el}(s_{j})(u_{j}(s_{j})) → E^{el}(t)(u^{∗}(t)).

By (7.5), comparing u_{j}(t) with u_{j}(s_{j}) − g_{j}(s_{j}) + g_{j}(t), taking into account that

S^{g_{j}(s_{j})}(u_{j}(s_{j})) ⊆ Γ_{j}(s_{j}) ⊆ Γ_{j}(t),

we obtain

E^{el}(t)(u_{j}(t)) ≤ E^{el}(s_{j})(u_{j}(s_{j})) + o_{j}

where o_{j} → 0 as j → +∞. Passing to the limit in j we have by (7.28)

E^{el}(t)(u(t)) ≤ lim inf_{j} E^{el}(t)(u_{j}(t)) ≤ lim sup_{j} E^{el}(t)(u_{j}(t)) ≤ E^{el}(t)(u^{∗}(t)).

By (7.27) we deduce that (7.17) holds. Moreover we have that E^{el}(t)(u(t)) = E^{el}(t)(u^{∗}(t)) and that E^{el}(t)(u(t)) is independent of the accumulation point u(t). Then we deduce that (7.18) holds.

Finally, concerning point (d), we have that for all s ∈ D ∩ [0, t[

lim inf_{n} E^{s}(Γ_{ε_{n},a_{n}}^{δ_{n}}(t)) ≥ lim inf_{n} E^{s}(Γ_{ε_{n},a_{n}}^{δ_{n}}(s)) ≥ E^{s}(Γ(s)),

so that letting s ↗ t we obtain (7.19). The proof is now complete. □

We can now prove Theorem 7.1.

\underline{PROOF OF THEOREM 7.1}. Let (δ_{n}, ε_{n}, a_{n})_{n∈ℕ} and \{t → Γ(t) ∈ R(\overline{Ω}_{B}; ∂_{N}Ω), t ∈ [0, T]\} be given by Lemma 7.4. For all t ∈ [0, T], let us set

u_{n}(t) := u_{ε_{n},a_{n}}^{δ_{n}}(t), Γ_{n}(t) := Γ_{ε_{n},a_{n}}^{δ_{n}}(t).

Let us see that it is possible to choose an accumulation point u(t) ∈ GSBV_{q}^{p}(Ω; ℝ^{2}) of (u_{n}(t))_{n∈ℕ} such that {t → (u(t), Γ(t)) : t ∈ [0, T]} is a quasistatic growth of brittle fractures in the sense of Dal Maso-Francfort-Toader. Let us set

ϑ_{n}(s) := ⟨∂W(∇u_{n}(s)), ∇ġ_{ε_{n}}(s)⟩

− Ḟ(s)(u_{n}(s)) − ⟨∂F(s)(u_{n}(s)), ġ_{ε_{n}}(s)⟩
− Ġ(s)(u_{n}(s)) − ⟨∂G(s)(u_{n}(s)), ġ_{ε_{n}}(s)⟩.

By growth conditions of W, F, G and by (7.4) we have that there exists ψ ∈ L^{1}(0, T) such that ϑ_{n}(s) ≤ ψ(s) for all n. Let us consider

ϑ(s) := lim sup_{n} ϑ_{n}(s).

By [13, Theorem 5.5 and Lemma 4.11], for every s ∈ [0, T] there exists u(s) accumulation point of (u_{n}(s))_{n∈ℕ} for the weak convergence in GSBV_{q}^{p}(Ω; ℝ^{2}) such that

ϑ(s) := ⟨∂W(∇u(s)), ∇ġ(s)⟩

− Ḟ(s)(u(s)) − ⟨∂F(s)(u(s)), ġ(s)⟩
− Ġ(s)(u(s)) − ⟨∂G(s)(u(s)), ġ(s)⟩.

Applying Fatou’s Lemma (in the limsup version) to (7.6) with s = 0, we have that

E(t)(u(t), Γ(t)) ≤ lim sup_{n} E(0)(u_{n}(0), Γ_{n}(0)) + ∫_{0}^{t} ϑ(s) ds.

By Proposition 5.1, we have that lim sup_{n} E(0)(u_{n}(0), Γ_{n}(0)) = E(0)(u(0), Γ(0)), so that we get

E(t)(u(t), Γ(t)) ≤ E(0)(u(0), Γ(0)) + ∫_{0}^{t} ϑ(s) ds.

Moreover, again by [13, Theorem 3.13],

E(t)(u(t), Γ(t)) ≥ E(0)(u(0), Γ(0)) + ∫_{0}^{t} ϑ(s) ds,

so that

(7.29)  E(t)(u(t), Γ(t)) = E(0)(u(0), Γ(0)) + ∫_{0}^{t} ϑ(s) ds.

We deduce that {t → (u(t), Γ(t)) : t ∈ [0, T]} is a quasistatic growth of brittle fractures: in fact by Lemma 7.4 we get that Γ(·) is increasing, and for t ∈ [0, T] (u(t), Γ(t)) ∈ AD(g(t)) and the static equilibrium holds; moreover the nondissipativity condition is given by (7.29).

Let us see that points (a) and (b) of Theorem 7.1 holds. By (7.4), (u_{n}(t))_{n∈ℕ} is weakly precompact in GSBV_{q}^{p}(Ω; ℝ^{2}) for all t ∈ [0, T]. Moreover by Lemma 7.4 every accumulation point ũ(t) of (u_{n}(t))_{n∈ℕ} for the weak convergence in GSBV_{q}^{p}(Ω; ℝ^{2}) is such that S^{g(t)}(ũ(t)) ⊆ Γ(t) and the minimality property (7.7) holds. Moreover we have

E^{el}(t)(ũ(t)) = lim_{n} E^{el}(t)(u_{n}(t)).

Since E^{el}(t)(ũ(t)) is independent of the particular accumulation point ũ(t), we have that point (a) is proved.

Let us come to point (b). Taking into account (7.18) and (7.19), for all t ∈ [0, T] we have

E(t) ≤ lim inf_{n} E_{n}(t) ≤ lim sup_{n} E_{n}(t) ≤ E(0) + ∫_{0}^{t} ϑ(s) ds = E(t),

so that (7.8) holds. Moreover we deduce that separate convergence of elastic and surface energies holds at any time, so that (7.9) is proved. The proof is now concluded. □

8. The strictly convex case

In this section we assume that the function W (x, ξ) is strictly convex in ξ for a.e. x ∈ Ω and that the function F (t, x, z) is strictly convex in z for all t ∈ [0, T] and for a.e. x ∈ Ω: as a consequence, the elastic energy E^{el}(t, v) is strictly convex in v for all t ∈ [0, T], and a stronger approximation result is available.

Theorem 8.1. Let g ∈ W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})) and let

g_{ε} ∈ W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})), g_{ε}(t) ∈ AF_{ε}(Ω; ℝ^{2}) for all t ∈ [0, T]

be such that for ε → 0

g_{ε} → g strongly in W^{1,1}([0, T], W^{1,p}(Ω; ℝ^{2}) ∩ L^{q}(Ω; ℝ^{2})).

Let Γ^{0} ∈ Γ(Ω) be an initial crack and let Γ_{ε,a}^{0} be its approximation in the sense of Proposition 5.1. Let us suppose that

(8.1)  W (x, ·) is strictly convex for a.e. x ∈ Ω,
F (t, x, ·) is strictly convex for a.e. (t, x) ∈ [0, T] × Ω.

Given δ > 0, ε > 0, a ∈]0, \frac{1}{2}[, let \{t → (u_{ε,a}^{δ}(t), Γ_{ε,a}^{δ}(t)) : t ∈ [0, T]\} be the piecewise constant interpolation of the discrete evolution given by Proposition 6.1 relative to the initial crack Γ_{ε,a}^{0} and the boundary data g_{ε}. Then there exists a quasistatic evolution {t → (u(t), Γ(t)) : t ∈ [0, T]} relative to the initial crack Γ^{0} and the boundary data g in the sense of Theorem 3.1, and sequences δ_{n} → 0, ε_{n} → 0, a_{n} → 0, such that setting

u_{n}(t) := u_{ε_{n},a_{n}}^{δ_{n}}(t), Γ_{n}(t) := Γ_{ε_{n},a_{n}}^{δ_{n}}(t),

for all t ∈ [0, T] the following facts hold:

(a) ∇u_{n}(t) → ∇u(t) strongly in L^{p}(Ω; M^{2×2}) and u_{n}(t) → u(t) strongly in L^{q}(Ω; ℝ^{2});

(b) E(t)(u_{n}(t), Γ_{n}(t)) → E(t)(u(t), Γ(t)), and in particular elastic and surface energies converge separately, that is

E^{el}(t)(u_{n}(t)) → E^{el}(t)(u(t)), E^{s}(Γ_{n}(t)) → E^{s}(Γ(t)).

Proof. Let us consider the sequence (δ_{n}, ε_{n}, a_{n})_{n∈ℕ} and the quasistatic growth of brittle fractures {t → (u(t), Γ(t)) : t ∈ [0, T]} given in Theorem 7.1. Under assumptions (8.1), we have that u(t) is uniquely determined, because by (7.7) u(t) minimizes

min{E^{el}(t)(v) : v ∈ GSBV_{q}^{p}(Ω; ℝ^{2}), S^{g(t)}(v) ⊆ Γ(t)},

and E^{el}(t)(·) is strictly convex. We conclude by point (a) of Theorem 7.1 that u_{n}(t) ⇀ u(t) weakly in GSBV_{q}^{p}(Ω; ℝ^{2}). Point (b) is a direct consequence of Theorem 7.1. By the convergence of the elastic energy, we deduce that

lim_{n} ∫_{Ω} W (x, ∇u_{n}(t)) dx = ∫_{Ω} W (x, ∇u(t)) dx,
lim_{n} ∫_{Ω} F (t, x, u_{n}(t)) dx = ∫_{Ω} F (t, x, u(t)) dx.

By the assumption on the strict convexity of W and F we deduce by [7]

∇u_{n}(t) → ∇u(t) strongly in L^{p}(Ω; M^{2×2}),

and

u_{n}(t) → u(t) strongly in L^{q}(Ω; ℝ^{2}).

Point (a) is now proved, and the proof is concluded. □

ACKNOWLEDGMENTS

The authors wish to thank Gianni Dal Maso for many helpful and interesting discussions.

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(Alessandro Giacomini) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy E-mail address, A. Giacomini: giacomin@sissa.it

(Marcello Ponsiglione) S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy E-mail address, M. Ponsiglione: ponsigli@sissa.it