Energy release rate for non smooth cracks in planar elasticity

This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.


Introduction
Griffith theory [18] is a model explaining the quasi-static crack growth in elastic bodies under the assumption that the crack set is preassigned. In a two-dimensional setting, let us denote by Ω ⊂ R 2 the reference configuration of a linearly elastic body allowing for cracks inside Γ. To fix the ideas, provided the evolution is sufficiently smooth, that Γ is a simple curve, and that the evolution is growing only in one direction, then the crack is completely characterized by the position of its tip, and thus by its arc length. Denoting by Γ( ) the crack of length inside Γ, the elastic energy associated to a given kinematically admissible displacement u : Ω Γ( ) → R 2 satisfying u = ψ(t) on ∂Ω Γ( ), is given by where C is the fourth order Hooke's tensor, e(u) is the symmetrized gradient of u, and ψ(t) : ∂Ω → R 2 is a prescribed boundary datum depending on time, which is the driving mechanism of the process. If the evolution is slow enough, it is reasonable to neglect inertia and viscous effects so that the quasi-static assumption becomes relevant: at each time t, the body is in elastic equilibrium. It enables one to define the potential energy as P(t, ) := E(t; u(t, ), ) = min E(t; ·, ), where the minimum is computed over all kinematically admissible displacements at time t. Therefore, given a cracking state, the quasi-static assumption permits to find the displacement. In order to get the crack itself (or equivalently its length), Griffith introduced a criterion whose fundamental ingredient is the energy release rate. It is defined as the variation of potential energy along an infinitesimal crack increment, or in other words, the quantity of released potential energy with respect to a small crack increment. More precisely, it is given by provided the previous expression makes sense. From a thermodynamical point of view, the energy release rate is nothing but the thermodynamic force associated to the crack length (the natural internal variable modeling the dissipative effect of fracture). Griffith's criterion is summarized into the three following items: for each t > 0 (i) G(t, (t)) G c , where G c > 0 is a characteristic material constant referred to as the toughness of the body; (ii)˙ (t) 0; (iii) G(t, (t)) − G c ˙ (t) = 0.
Item (i) is a threshold criterion which stipulates that the energy release rate cannot exceed the critical value G c . Item (ii) is an irreversibility criterion which ensures that the crack can only grow. The third and last item is a compatibility condition between (i) and (ii): it states that a crack will grow if and only if the energy release rate constraint is saturated.
An existence result for this model has been given in [5] (see also [13,16,12] in other contexts) for cracks belonging to the class of compact and connected subsets of Ω.
The main reason of this assumption was to ensure the lower semicontinuity of the Mumford-Shah type functional (u, Γ) → E (u, Γ) with respect to a reasonable notion of convergence. The lower semicontinuity of the surface energy with respect to the Hausdorff convergence of cracks is a consequence of Gołab's Theorem (see [15]), while the continuity of the bulk energy is a consequence of continuity results of the Neumann problem with respect to the Hausdorff convergence of the boundary (see [4,6]) together with a density result [5]. In any cases, all these results only hold in dimension 2 and in the class of compact and connected sets. If one is interested into fine qualitative results such as crack initiation (see [8]) of kinking (see [7]) it becomes necessary to understand the nature of the singularity at the crack tip. Therefore one should be able to make rigorous a suitable notion energy release rate. The first proof of the differentiable character of the potential energy with respect to the crack length has been given in [14] (see also [22,28,27]). The generalized variational setting described above, a mathematical justification of the notions of energy release rate for any incremental crack attached to a given initial crack has been in [7] in the case where the crack is straight in a small neighborhood of its tip. In the footstep of that work, we attempt here weaken the regularity assumption on the initial crack, which is merely closed, connected, with density 1/2 at the origin (that imply to blow up as a segment at the origin, up to rotations).

Main
Results. -Our main results are contained in Theorem 6.4 and Theorem 7.1 respectively in Section 6 and Section 7.
The first main result Theorem 6.4 is a purely P.D.E. result. We analyze the blow-up limit of the optimal displacement at the tip of the given initial crack. We prove that for some suitable subsequence, the blow-up limit converges to the classical crack-tip function in the complement of a half-line, i.e., of the form for some constants κ 1 and κ 2 ∈ R, while φ 1 and φ 2 are positively 1/2-homogenous functions which are explicitly given by (6.15) and (6.16) below. This part can be seen as a partial generalization in planar elasticity of what was previously done in the anti-plane case [9]. Mathematically speaking, the corresponding function to be studied is now a vectorial function satisfying a Lamé type system, instead of being simply a scalar valued harmonic function. One of the key obstacles in the vectorial case is that no monotonicity property is known for such a problem, which leads to a slightly weaker result than in the scalar case: the convergence of the blow-up sequence only holds up to subsequences, and nothing is known for the whole sequence. Consequently, the constants κ 1 and κ 2 in (1.1) a priori depend on this particular subsequence. As a matter of fact, this prevents us to define properly the stress intensity factor analogously to what was proposed in [9]. On the other hand, we believe that the techniques employed in the proof and the results on their own are already interesting. In addition, the absence of monotonicity is not the only difference with the scalar case, which led us to find a new proof relying on a duality approach via the so-called Airy function in order to bypass some technical problems.
Another substantial difference with the scalar case appears while studying homogeneous solutions of the planar Lamé system in the complement of a half-line, which is crucial in the understanding of blow-up solutions at the crack tip. For harmonic functions it is rather easy to decompose any solutions as a sum of spherical-harmonics directly by writing the operator in polar coordinates, and identify the degree of homogeneity of each term with the corresponding eigenvalue of the Dirichlet-Laplace-Beltrami operator on the circle minus a point. For the Lamé system, or alternatively for the biharmonic equation, a similar naive approach cannot work. The appropriate eigenvalue problem on the circle have a more complicate nature, and analogous results rely on an abstract theory developed first by Kondrat'ev which rests on pencil operators, weighted Sobolev spaces, the Fredholm alternative, and calculus of residues. We used this technology in the proof of Proposition 6.3 for which we could not find a more elementary argument.

Second result. -
The second main result Theorem 7.1 concerns the energy release rate of an incremental crack Γ, which is roughly speaking the derivative of the elastic energy with respect to the crack increment (see (7.1) for the precise definition). We prove that the value of this limit is realized as an explicit minimization problem in the cracked-plane R 2 (−∞, 0]×{0} . One can find a similar statement in [7, Th. 3.1], but with the additional assumption that the initial crack is a line segment close to the origin. We remove here this hypothesis, establishing the same result for any initial crack which is closed, connected and admits a line segment as blow-up limit at the origin. The starting point for this generalization is the knowledge of the blow-up limit at the origin for displacement associated to a general initial crack, namely our first result Theorem 6.4. Since this result holds only up to subsequences, the same restriction appears in the statement of Theorem 6.4 as well.
Therewith, it should be mentioned that Theorem 7.1 is new even for the scalar case, for which the conclusion is even more accurate. Indeed in this case, the monotonicity formula of [9] ensures that the convergence holds for the whole sequence and not only for a subsequence.
The paper is organized as follows: after introducing the main notation in Section 2, we describe precisely the mechanical model in Section 3. Section 4 is devoted to establish technical results related to the existence of the harmonic conjugate and the Airy function associated to the displacement in a neighborhood of the crack tip. In Section 5, we prove lower and upper bounds of the energy release rate. The blow-up analysis of the displacement around the crack tip is the object of Section 6. Section 7 is devoted to give a formula for the energy release rate as a global minimization problem. Finally, we shortly review Kondrat'ev theory of elliptic regularity vs singularity inside corner domains in an appendix.

Mathematical preliminaries
2.1. General notation. -The Lebesgue measure in R n is denoted by L n , and the k-dimensional Hausdorff measure by H k . If E is a measurable set, we will sometimes write |E| instead of L n (E). If a and b ∈ R n , we write a · b = n i=1 a i b i for the Euclidean scalar product, and we denote the norm by |a| = √ a · a. The open ball of center x and radius is denoted by B (x). If x = 0, we simply write B instead of B (0).
We write M n×n for the set of real n × n matrices, and M n×n sym for that of all real symmetric n × n matrices. Given a matrix A ∈ M n×n , we let |A| := tr(AA T ) (A T is the transpose of A, and tr A is its trace) which defines the usual Euclidean norm over M n×n . We recall that for any two vectors a and b ∈ R n , a ⊗ b ∈ M n×n stands for the tensor product, i.e., (a ⊗ b) ij = a i b j for all 1 i, j n, and a b : sym denotes the symmetric tensor product. Given an open subset U of R n , we denote by M (U ) the space of all real-valued Radon measures with finite total variation. We use standard notation for Lebesgue spaces L p (U ) and Sobolev spaces W k,

2.2.
Capacities. -In the sequel, we will use the notion of capacity for which we refer to [1,21]. We just recall the definition and several facts. The (k, 2)-capacity of a compact set K ⊂ R n is defined by This definition is then extended to open sets A ⊂ R n by One of the interests of capacity is that it enables one to give an accurate sense to the pointwise value of Sobolev functions. More precisely, every u ∈ H k (R n ) has a (k, 2)-quasicontinuous representative u, which means that u = u a.e. and that, for each ε > 0, there exists a closed set A ε ⊂ R n such that Cap k,2 (R n A ε ) < ε and u| Aε is continuous on A ε (see [1, Sec. 6.1]). The (k, 2)-quasicontinuous representative is unique, in the sense that two (k, 2)-quasicontinuous representatives of the same function u ∈ H k (R n ) coincide Cap k,2 -quasi-everywhere. In addition, if U is an open subset of R n , then u ∈ H k 0 (U ) if and only if for all multi-index α ∈ N n with length |α| k, ∂ α u has a (k − |α|, 2)-quasicontinuous representative that vanishes Cap k−|α|,2 -quasi everywhere on ∂U , i.e., outside a set of zero Cap k−|α|,2 -capacity (see [1,Th. 9.1.3]). In the sequel, we will only be interested to the cases k = 1 or k = 2 in dimension n = 2.
It will also be useful to introduce the spaces V β (C) for < 0, which is defined as the dual space of V − −β (C), endowed with the usual dual norm. Observe that when 0 then u ∈ V β (C) if and only if the function x → |x| β− +|α| ∂ α u(x) ∈ L 2 (C) for all |α| . If one is interested in homogeneous functions, it turns out that the parameter β plays a different role regarding to the integrability at the origin or at infinity. To fix the ideas, one can check that in dimension 2, a function of the form x → |x| γ f (x/|x|) around the origin and with compact support belongs to V β (C) for every β < 1 − γ. On the other hand, a function having this behavior at infinity and vanishing around the origin will belong to a space V β (C) for every β > 1 − γ. For instance if γ = 3/2, then the corresponding space of critical exponent would be that with β = −1/2. In linearized elasticity, u stands for the displacement, while e(u) is the elastic strain. The elastic energy of a body is given by a quadratic form of e(u) so that it is natural to consider displacements such that e(u) ∈ L 2 (U ; M n×n sym ). If U has Lipschitz boundary, it is well known that u actually belongs to H 1 (U ; R n ) as a consequence of Korn's inequality (see e.g. [10,31]). However, when U is not smooth, we can only assert that u ∈ L 2 loc (U ; R n ). This motivates the following definition of the space of Lebesgue deformations: LD(U ) := {u ∈ L 2 loc (U ; R n ) : e(u) ∈ L 2 (U ; M n×n sym )}. If U is connected and u is a distribution with e(u) = 0, then necessarily it is a rigid movement, i.e., u(x) = Ax + b for all x ∈ U , for some skew-symmetric matrix A ∈ M n×n and some vector b ∈ R n . If, in addition, U has Lipschitz boundary, the following Poincaré-Korn inequality holds: there exists a constant c U > 0 and a rigid movement r U such that According to [2, Th. 5.2, Exam. 5.3], it is possible to make r U more explicit in the following way: consider a measurable subset E of U with |E| > 0, then one can take provided the constant c U in (2.1) also depends on E.

2.5.
Hausdorff convergence of compact sets. -Let K 1 and K 2 be compact subsets of a common compact set K ⊂ R n . The Hausdorff distance between K 1 and K 2 is given by We say that a sequence (K n ) of compact subsets of K converges in the Hausdorff distance to the compact set K ∞ if d H (K n , K ∞ ) → 0. The Hausdorff convergence of compact sets turns out to be equivalent to the convergence in the sense of Kuratowski. Indeed K n → K ∞ in the Hausdorff metric if and only if both following properties hold: a) any x ∈ K ∞ is the limit of a sequence (x n ) with x n ∈ K n ; b) if for all n, x n ∈ K n , any limit point of (x n ) belongs to K ∞ .
Finally let us recall Blaschke's selection principle which asserts that from any bounded sequence (K n ) of compact subsets of K, one can extract a subsequence converging in the Hausdorff distance.

2.6.
Convention. -If f : R m → R m is a vector field, the notation ∇f stands for the m × n matrix whose entries are ∂f i /∂x j (for 1 i m and 1 j n). In particular, if m = 1, i.e., if f is scalar valued, ∇f is a row vector. If n = 2 and m = 1, we write while if n = 2 and m = 2, we write

Description of the model
Reference configuration. -We consider a homogeneous isotropic linearly elastic body occupying Ω in its reference configuration, a bounded and connected open subset of R 2 with Lipschitz boundary. We suppose that the stress σ ∈ M 2×2 sym is related to the strain e ∈ M 2×2 sym thanks to Hooke's law where λ, µ ∈ R are the Lamé coefficients satisfying λ + µ > 0 and µ > 0, and I is the identity matrix. This expression can be inverted into where E := µ(3λ + 2µ)/(λ + µ) is the Young modulus and ν := λ/2(λ + µ) is the Poisson coefficient.
External loads. -We suppose that the body is only subjected to a soft device loading, that is, to a prescribed displacement ψ ∈ H 1/2 (∂Ω; R 2 ) acting on the entire boundary.
-We further assume that the body can undergo cracks which belong to the admissible class Admissible displacements. -For a given crack Γ ∈ K (Ω), we define the space of admissible displacement by has Lipschitz boundary so that Korn's inequality ensures that u ∈ H 1 (Ω ∩ B; R 2 ). As a consequence, the trace of u is well defined on ∂Ω ∩ B. Since this property holds for any ball as above, then the trace of u is well defined on ∂Ω Γ.
and an associated displacement u 0 ∈ LD(Ω Γ 0 ) given as a solution of the minimization problem Note that u 0 is unique up to an additive rigid movement in each connected component of Ω Γ 0 disjoint from ∂Ω Γ 0 . However, the stress, which is given by Hooke's law is unique and it satisfies the variational formulation Note that standard results on elliptic regularity (see e.g. [10, Th. 6.3.6]) ensure that u 0 ∈ C ∞ (Ω Γ 0 ; R 2 ).
Energy release rate. -To define the energy release rate, let us consider a crack incre- We define

Construction of dual functions
The goal of this section is to construct the harmonic conjugate and the Airy function associated to the displacement u 0 in a neighborhood of the crack tip which is assumed to be the origin. Their construction rests on an abstract functional analysis result (Lemma 4.1 below) which puts in duality gradients and functions with vanishing divergence outside a (non-smooth) crack.
Let B = B R0 and B = B R 0 be open balls centered at the origin with radii R 0 < R 0 , such that B ⊂ Ω and ∂B ∩ Γ 0 = ∅. By assumption, since Γ 0 ∈ K (Ω) satisfies (3.2), this property certainly holds true provided R 0 is small enough. Note in particular that the connectedness of Γ 0 ensures that ∂B ∩ Γ 0 = ∅ as well.
The following result is a generalization of [5, Lem. 1].
-Consider the following subspaces of L 2 (B; R 2 ): It follows that the product σv belongs to H 1 0 (B; R 2 ) and that (4.1) div(σv) = σ∇v, Consequently, X ⊂ Y ⊥ , and thus X ⊂ Y ⊥ . We next establish the reverse inclusion. Let Ψ ∈ X ⊥ . According to De Rham's Theorem (see [30, p. 20] then the open set V ∩ B has Lipschitz boundary, and thus v ∈ H 1 (V ∩ B). Therefore, the trace of v on ∂B Γ 0 is well defined, and we claim that we can assume that v = 0 on ∂B Γ 0 .
To show this property, we consider a connected component U of B Γ 0 , and let S = ∂B ∩ U . For any g ∈ C ∞ c (S) with zero average, it is possible to find a smooth, bounded and connected open set U ⊂ U such that both U 1 := U ∩B and U 2 := U B are connected and have Lipschitz boundary, and with Supp g ⊂ U ∩ ∂B. Next, we consider a smooth function with compact support in U , which coincides with g on S = S ∩ U (and which we still denote g ∈ C ∞ c (U )). We then denote by H 1 the subspace of H 1 (U 1 ) made of all functions ϕ ∈ H 1 (U 1 ) satisfying U1 ϕ dx = 0, and introduce the following variational problem: It is standard that this problem has a unique minimizer Since g ∈ C ∞ c (S ) and S is smooth, we deduce that ϕ 1 ∈ C ∞ (U 1 ∪ (S ∩ ∂U 1 )) by elliptic regularity. In addition, since for any ϕ ∈ H 1 0 (B), Next, we argue similarly in U 2 to get a function Since the normal trace of σ does not jump across S , σ is divergence free in H −1 (R 2 ). Therefore, taking a standard sequence of mollifiers (ρ ε ) ε>0 , and considering σ ε = σ * ρ ε | B , we get an element of X, and since Ψ ∈ X ⊥ we infer that and since g was arbitrary (with zero average on S) we deduce that v is a constant on S. Since S = ∂B ∩U and U is connected, we can remove this constant from v in the component U of B Γ 0 and assume that v = 0 on S. Doing the same in all connected components yields a function v which vanishes on ∂B Γ 0 . Finally, considering the truncated function v k : 4.1. The harmonic conjugate. -We are now in position to construct the harmonic conjugate v 0 associated to u 0 in B. By construction, the displacement u 0 satisfies a Neumann condition on the crack Γ 0 , while its associated stress σ 0 has zero divergence outside the crack, both in a weak sense. The harmonic conjugate v 0 is, roughly speaking, a dual function of u 0 in the sense that it satisfies a homogeneous Dirichlet boundary condition on the crack Γ 0 , and its rotated gradient coincides with the stress σ 0 . The harmonic conjugate will be of use in the proof of Proposition 5.1 in order to prove a lower bound on the energy release rate. It will also appear in the construction of the Airy function.

4.2.
The Airy function. -We next construct the Airy function w 0 associated to the displacement u 0 in B following an approach similar to [5]. This new function has the property to be a biharmonic function vanishing on the crack. Therefore, the original elasticity problem (3.3) can be recast into a suitable biharmonic equation whose associated natural energy (the L 2 norm of the Hessian) coincides with the original elastic energy. The Airy function will be useful in Section 6 in order to get an a priori bound on the rescaled elastic energy around the crack tip, as well as in our convergence result for the blow-up displacement. and Proof. -We reproduce the construction initiated in the proof of Proposition 4.2 with the larger ball B instead of B. It ensures the existence of p (1) and By definition, there exists sequences (p we infer thanks to the integration by parts formula that according again to Lemma 4.1. Arguing as in the proof of Proposition 4.2, we deduce the existence of some w 0 ∈ H 1 0,Γ0 (B ) such that ∇w 0 = (p (1) , p (2) ).
By construction, the Airy function w 0 satisfies (4.4). Consequently, we have Observe that since w 0 ∈ H 2 (B ), we can assume that it is continuous (that is, we consider its continuous representative), and even in C 0,α (B) for any α < 1.
Let us show that w 0 ∈ H 2 0,Γ0 (B). This property rests on a capacity argument similar to that used in [5,Th. 1]. Thanks to [1, Th. 9.1.3], we just need to check that w 0 vanishes on Γ 0 ∩ B, pointwise. First, we consider a cut-off function η ∈ C ∞ c (B ; [0, 1]) satisfying η = 1 on B. Denoting z 0 := ηw 0 , then one has and consider γ ⊂ K a connected component. Then the diameter of γ must be zero, otherwise γ would have positive Cap 1,2 -capacity (see [21,Cor. 3.3.25]). Therefore it must be at most an isolated point of Γ 0 . However, ∂(B Γ 0 ) is connected and therefore has no isolated point, hence K = ∅.
We next show that w 0 is a biharmonic function. Indeed, according to (4.4), one has Denoting by e 0 := e(u 0 ) the elastic strain, and using the compatibility condition together with Hooke's law (3.1), we infer that Remark 4.5. -According to the results of [24], we get the following estimate of the energy of w 0 around the origin: for every 2 < R R 0 , for some universal constant C 0 > 0 independent of R and . Indeed, it suffices to apply [24,Th. 2] in the open set B Γ 0 with (in their notation) ω = 2π and δ = 1/2. This is possible since, Γ 0 being connected, then for all < R we have ∂B ∩ Γ 0 = ∅, Thanks to the reformulation of the elasticity problem as a biharmonic equation, and according to Remark 4.5 concerning the behavior of the energy of a biharmonic function in fractured domains, we get the following result about the elastic energy concentration around the crack tip. We observe that in [9] a stronger result has been obtained in the scalar (anti-plane) case where a monotonicity formula has been established.
Proposition 4.6. -Let σ 0 be the stress defined in (3.4) and R 0 > 0 be such that B R0 ⊂ Ω and ∂B R0 ∩ Γ = ∅. Then there exists a universal constant C 0 > 0 such that for all ρ, R > 0 satisfying 2 < R R 0 , Proof. -The result is an immediate consequence of (4.4) together with Remark 4.5.

Bounds on the energy release rate
The goal of this section is to establish bounds on the energy release rate. This is the first step toward a more precise analysis and a characterization of the energy release rate as a limiting minimization problem (see Section 7). As in [7,Lem. 2.4], the proof of the upper bound relies on the construction of an explicit competitor for the minimization problem (3.7) defining G ε . The lower bound rests in turn into a dual formulation (in term in the stress) of the minimization problem (3.6), and into the construction, for each crack increment, of an admissible stress competitor for this new dual variational problem. The construction we use is based on the harmonic conjugate v 0 associated to the displacement obtained in Proposition 4.2.
This set clearly belongs to K (Ω) and We then apply Proposition 4.6 which shows that for some G * > 0.

Blow-up limit of the pre-existing crack
In this section we investigate the nature of the singularity of the displacement u 0 and the stress σ 0 at the origin, which is the tip of the crack Γ 0 having density 1/2 at that point. We will prove, that along suitable subsequences of radius ε k → 0 of balls, the rescaled crack converges in the Hausdorff sense to a half-line (modulo a rotation), and the rescaled displacement converges in a certain sense to the usual crack-tip function in the complement of a half-line. Once again, the analysis strongly relies on the Airy function introduced in Proposition 4.3. Contrary to [9] where the scalar antiplane was treated, we do not have any monotonicity formula on the energy (neither for the elastic problem nor for the biharmonic one) which prevents one to ensure the existence of the limit of the rescaled energy, and thus the uniqueness of the limit. Therefore, in contrast with [9], our result strongly depends upon the sequence (ε n ).

6.1.
Blow-up analysis of the Airy function.
-We first show that the Airy function blows-up into a biharmonic function outside the half line limit crack, satisfying a homogeneous Dirichlet condition on the crack, and that its energy computed on a ball behaves like the radius.
. In addition, w Σ0 is a solution of the following biharmonic problem with homogeneous Dirichlet boundary condition on the crack: for any R > 0, and it satisfies the following energy bound Proof. -The proof is divided into several steps. We first derive weak compactness on the rescaled Airy function, according the energy bound of the original Airy function. We then derive a Dirichlet condition on the crack for the weak limit and its gradient. Using a cut-off function argument, we establish that the weak convergence is actually strong, which enables one to show that the limit Airy function is a biharmonic function outside the crack. In the sequel R > 0 is fixed, and ε > 0 is small enough such that 2R < R 0 /ε.
Strong convergence. -Our aim now is to prove that w ε k → w Σ0 strongly in H 2 loc (R 2 ). By the lower semicontinuity of the norm with respect to weak convergence, we already have for any r < 2R so that it is enough to prove the converse inequality with a lim sup. To this aim we will use the minimality property of w ε k , and suitably modify w Σ0 into an admissible competitor. Let us select a radius r ∈ (R, 2R) such that µ(∂B r ) = 0. Since w Σ0 ∈ H 2 0,Σ0 (B r ), for every n ∈ N, there exists a function h n ∈ C ∞ (B r ) such that Supp(h n ) ∩ Σ 0 = ∅ and h n → w Σ0 in H 2 (B r ) as n → ∞. Note that, by Hausdorff convergence, one also has that Supp(h n ) ∩ Σ ε k = ∅ for k k n large enough, for some integer k n ∈ N.
Let us consider a cut-off function η δ ∈ C ∞ c (B r ; [0, 1]) satisfying We finally define Observe that z δ,n,k ∈ H 2 0,Σε k (B r ) provided that k k n is large enough. Consequently, since z δ,n,k ∈ w ε k + H 2 0 (B r ), we infer thanks to (4.3) and Remark 4.4 that J.É.P. -M., 2015, tome 2 By convexity, we get that and thanks to (6.8) Letting first k → ∞ and then n → ∞, using that w ε k → w Σ0 in H 1 (B r ) and that On the other hand Therefore we can write that lim sup k→∞ Br Finally, letting δ → 0 in (6.9) and using the fact that µ(∂B r ) = 0, we get the desired bound lim sup k→∞ Br which ensures together with (6.7) that w ε k converges strongly to w Σ0 in H 2 (B r ).
Biharmonicity. -In order to show that w Σ0 solves a biharmonic Dirichlet problem outside the crack Σ 0 is is enough to check that it satisfies the minimality property . Since z n = 0 in a neighborhood of Σ 0 , it follows by Hausdorff convergence that z n = 0 in a neighborhood of Σ ε k for k k n large enough, for some integer k n ∈ N. Therefore, for any k k n , w ε k + z n ∈ w ε k + H 2 0,Σε k (B R ) is an admissible competitor for the minimality property satisfied by the Airy function (see Remark 4.4), and Letting first k → ∞ and then n → ∞, and using the strong convergence of (w ε k ) established before yields The proof of the proposition is now complete.
Remark 6.2. -By elliptic regularity, it follows that w Σ0 is smooth outside the origin up to both sides of Σ 0 . In particular, for every 0 < r < R < ∞ and for every k ∈ N, w Σ0 ∈ H k ((B R B r ) Σ 0 ) and is a solution for problem (6.4) in a stronger sense.
It turns out that w Σ0 can be made explicit by showing that it is a positively 3/2homogeneous function. The proof of this result follows an argument given by Monique Dauge, relying on the theory introduced by Kondrat'ev in [23], that is briefly recalled in the appendix. Proof. -Let w Σ0 be the biharmonic function in R 2 Σ 0 with homogeneous Dirichlet boundary conditions given by Proposition 6.1, and let χ ∈ C ∞ c (R 2 ; [0, 1]) be a cut-off function satisfying χ = 1 in B 1 and χ = 0 in R 2 B 2 . We decompose w Σ0 as follows: where w 0 := χw Σ0 and w ∞ := (1 − χ)w Σ0 . Of course both w 0 and w ∞ still satisfy homogenous boundary Dirichlet conditions on Σ 0 , and one can check that for some f 0 and f ∞ supported in the annulus B 2 B 1 . In addition, according to Remark 6.2, it follows that both f 0 and f ∞ ∈ H k (R 2 Σ 0 ) for every k ∈ N, and consequently f 0 and f ∞ ∈ V β (R 2 Σ 0 ) for all ∈ Z and all β ∈ R (we recall Section 2.3 for the definition of V β ). We next intend to apply Theorem A.2 to w 0 and w ∞ separately.

6.2.
Blow-up analysis of the displacement. -We are now in position to study the blow-up of the displacement. We show that, up to a subsequence and rigid movement, it converges to the usual positively 1/2-homogeneous function satisfying the Lamé system outside a half-line.
Theorem 6.4. -For every sequence (ε n ) 0 + , there exist a subsequence (ε k ) ≡ (ε n k ) 0 + , a sequence (m k ) of rigid movements and a function u Σ0 ∈ LD loc (R 2 Σ 0 ) such that the blow-up sequence of displacements satisfies . In addition, the function u Σ0 is positively 1/2-homogeneous and it is given in polar coordinates by where κ 1 and κ 2 ∈ R are constants, while φ 1 and φ 2 are defined by Proof. -A scalar version of that theorem is contained in [9, Th. 1.1], but the proof does not extend directly to the vectorial case. This is why we present here an alternative argument based on the Airy function. Let (ε k ) be the subsequence given by Proposition 6.1. As in the proof of that result, R > 1 is fixed, and k ∈ N is large enough such that 2R < R 0 /ε k .
-We next show that u Σ0 satisfies the minimality property Moreover, since {0} has zero Cap 1,2 -capacity, we can also assume without loss of generality that v = 0 in a neighborhood of the origin.
. Therefore, thanks to the minimality property (3.3) satisfied by u 0 , we infer that so that passing to the limit as k → ∞, and invoking the strong convergences (6.13) yields the desired minimality property.
The previous expression of the displacement shows that where Φ is a positively −1/2-homogeneous function. On the other hand, passing to the limit in (6.17) as k → ∞ and using Proposition 6.1 yields å .
According to Proposition 6.3 the right hand side of the previous equality is positively −1/2-homogeneous as well. Therefore gathering (6.19) and (6.20) ensures that e(g) = 0 which shows that g = m is a rigid movement. We finally define the rigid displacement m k := u k + m which fulfills the conclusions of the proposition.

Energy release rate
Following the approach of [7], our aim is to give a definition of energy release rate by studying the convergence of the blow-up functional 1 ε G (εΓ). The following statement is the same as [7, Th. 3.1], but with the substantial difference that now Γ 0 is not assumed to be a straight line segment near the origin, but only blowing-up to such a segment for the Hausdorff distance.
Theorem 7.1. -Let (Γ ε ) ε>0 be a sequence of crack increments in K (Ω) be such that sup ε H 1 (Γ ε ) < ∞, and Γ ε → Γ in the sense of Hausdorff in Ω. Let us consider the rescaled crack Σ ε and displacement u ε defined, respectively by (6.1) and (6.2). Then for every sequence (ε n ) 0 + , there exist a subsequence (ε k ) ≡ (ε n k ) 0 + and a rotation R ∈ SO(2) such that where F is defined by where R > 0 is any radius such that Γ ⊂ B R , and u Σ0 is the function introduced in Theorem 6.4.
Remark 7.2. -The proof of Theorem 7.1 follows the scheme of [7, Th. 3.1], but some technical issues arise at two main points: 1) the explicit expression for the blow-up at the origin does not come directly from the literature but now follows from our first main result Theorem 6.4, and 2) the construction of a recovery sequence of functions in the moving domains that converges in a strong sense to prove the minimality of the limit is more involved, since now after rescaling everything in B 1 our sequence of domains also moves on ∂B 1 .
Remark 7.3. -In the scalar case (antiplane) the limit does actually not depend on the subsequence due to the existence of blow-up limit for the whole sequence [9].
Proof of Theorem 7.1. -Let (ε n ) 0 + and (ε k ) ≡ (ε n k ) ⊂ (ε n ) be the subsequence given by Theorem 6.4. Let us consider the rotation R ε be introduced at the beginning of Section 6. It is not restrictive to assume that R ε k converges to some limit rotation R. In particular R ε k (Γ ε k ) converges to R(Γ) in the sense of Hausdorff.
Rescaling. -We denote by u k a solution of the minimization problem and Recalling (3.6) and (3.7), we can write and setting w k := u k − u 0 , we obtain that Since w k = 0 on ∂Ω (Γ 0 ∪ ε k Γ ε k ), the variational formulation of (7.3) ensures that Ω Ce(u k ) : e( w k ) dx = 0, and it follows, writing u 0 = u k − w k , On the other hand, from (7.3) it is easy to see that 1 ε k G (ε k Γ ε k ) is also resulting from a minimization problem with homogeneous boundary condition. Indeed, for any w ∈ LD(Ω (Γ 0 ∪ ε k Γ ε k )) with w = 0 on ∂Ω (Γ 0 ∪ ε k Γ ε k ), denoting v = u 0 + w, we obtain that which implies According to the assumptions done on Γ ε , there exists R > 0 such that if ε is small enough, then Γ ε ⊂ B R ⊂ Ω, and H 1 (εΓ ε ) Cε for some constant C > 0 independent of ε. In addition, thanks to the lower bound in Proposition 5.1, we get again for ε small enough, which implies from (7.4) We now proceed to the following change of variable: . We easily deduce from (7.6) that We can also recast the minimization problem in (7.5) in terms of w k , which now writes as where we used (7.4) in the last equality. Compactness.
-We now extend w k by 0 outside Ω k in such a way that w k ∈ LD(R 2 (Σ ε k ∪ R ε k (Γ ε k ))). Defining e k := e(w k ) in Ω k 0 otherwise, and using (7.7) together with the coercivity of C, we infer that the sequence (e k ) k∈N is uniformly bounded in L 2 (R 2 ; M 2×2 sym ). Consequently, up to a new subsequence (not relabeled), we can assume that e k e weakly in L 2 (R 2 ; M 2×2 sym ) for some function e ∈ L 2 (R 2 ; M 2×2 sym ). Let us recall that Σ ε k → Σ 0 := (−∞, 0] × {0} locally in the sense of Hausdorff in R 2 , and that Γ ε k → Γ in the sense of Hausdorff in Ω. Let us denote by " B := B 1/2 ((R + 1, 0)) the ball of R 2 centered at the point (R + 1, 0) and of radius 1/2. Since Γ ⊂ B R and thus R(Γ) ⊂ B R , we deduce that (Σ 0 ∪ R(Γ)) ∩ " B = ∅. Therefore, for k large enough, " ). Let us consider a bounded and smooth open set U ⊂ R 2 (Σ 0 ∪ R(Γ)) containing " B. Then for all k large enough, we have U ⊂ Ω k (Σ ε k ∪ R ε k (Γ ε k )), and we denote by r k the rigid movement defined by By Korn's inequality, we obtain that for some constant C U > 0 depending on U but independent of k. This implies that, up to a subsequence, w k − r k w weakly in H 1 (U ; R 2 ) for some w ∈ H 1 (U ; R 2 ). By exhausting R 2 (Σ 0 ∪ R(Γ)) with countably many open sets, extracting successively many subsequences and using a diagonal argument, we obtain that w ∈ H 1 loc (R 2 (Σ 0 ∪ R(Γ)); R 2 ) and w k − r k w weakly in H 1 loc (R 2 (Σ 0 ∪ R(Γ)); R 2 ).
Lower bound inequality. -Let ζ ∈ W 1,∞ (R 2 ; [0, 1]) be a cut-off function such that ζ = 1 on B R and ζ = 0 on R 2 B R for some given R > R. Recalling (7.8) we can write Since Γ ε k ⊂ B R , then (1−ζ)w k +ζr k = r k in B R . On the other hand, (1−ζ)w k +ζr k = w k = 0 on ∂Ω k Σ ε k so that it is and admissible variation for the minimization problem defining u ε k . We deduce and it follows Recalling from Theorem 6.4 that u ε k → u Σ0 strongly in L 2 loc (R 2 ; R 2 ), and e(u ε k ) → e(u Σ0 ) strongly in L 2 loc (R 2 ; M 2×2 sym ), while w k − r k → w strongly in L 2 loc (R 2 ; R 2 ), and e(w k − r k ) e(w) weakly in L 2 (R 2 ; M 2×2 sym ), we infer that We now let ζ be the Lipschitz and radial function defined by Letting R → R in the right-hand side of (7.9) we finally get that, for L 1 -a.e. R > 0, Reduction to competitors in H 1 (R 2 (Σ 0 ∪ R(Γ)); R 2 ) with compact support. -In order to show that w is a minimizer of the limit problem (7.2), we start by establishing that, without loss of generality, competitors in (7.2) can be taken in H 1 (R 2 (Σ 0 ∪ R(Γ)); R 2 ) with compact support. First we reduce to the case where the competitor belong to LD(R 2 (Σ 0 ∪ R(Γ))) have compact support. To this purpose, let us show that any z ∈ LD(R 2 (Σ 0 ∪ R(Γ))) can be approximated strongly in LD(R 2 (Σ 0 ∪ R(Γ))) by functions with compact support. To this aim we consider ϕ ∈ C ∞ c (B 2 ; [0, 1]) satisfying ϕ = 1 on B 1 , and define ϕ R (x) := ϕ (x/R) .
We assume that R is large enough so that Γ ⊂ B R . Then we set z R := (z − m R )ϕ R where m R is a suitable rigid movement associated to the Poincaré-Korn inequality in the domain B 2R (B R ∪ Σ 0 ) (note that this domain is not Lipschitz but it is a connected finite union of Lipschitz domains in which Korn's inequality remains valid), namely Moreover a immediate computation yields The first term converges strongly to e(z) in L 2 (R 2 ; M 2×2 sym ), while the second term converges to 0 strongly in L 2 (R 2 ; M 2×2 sym ) due to (7.11). As a consequence z R → z strongly in LD(R 2 (Σ 0 ∪ R(Γ))).
Next, we reduce to the case where z lies in the Sobolev space Let D and D be bounded open sets such that Supp(z) ⊂ D D. According to the density result [5, Th. 1], we get the existence of a sequence such that z n → z strongly in L 2 (D; R 2 ) and e(z n ) → e(z) both strongly in L 2 (D; M 2×2 sym ). This implies in particular that z n → 0 in L 2 (D D ; R 2 ). Let ϕ ∈ C ∞ c (D; [0, 1]), ϕ = 1 on D , and set z n = ϕz n ∈ H 1 (R 2 (Σ 0 ∪ R(Γ))) with Supp( z n ) ⊂ D, and satisfying z n → z strongly in L 2 (R 2 ; R 2 ), and e( z n ) → e(z) strongly in L 2 (R 2 ; M 2×2 sym ).
Let ζ be the cut-off function defined in (7.10), then performing an integration by parts exactly as we did in step 3 (with z k instead of w k − r k ) we arrive at the following The convergences established so far for the sequences (z k ) and (u ε k ) enable one to pass to the limit in the previous expression, first as k → ∞ and then R → R. We finally get that (7.13) lim sup Ce(u Σ0 ) : e(z) dx + ∂B R z · (Ce(u Σ0 )ν) dH 1 for almost every R > 0. By the density result established in step 4, inequality (7.13) holds for any z ∈ LD(R 2 (Σ 0 ∪ R(Γ))). Taking z = w, and gathering with (7.9) yields lim j→∞ 1 ε k G (ε k Γ ε k ) = 1 2 R 2 Ce(w) : e(w) dx Ce(u Σ0 ) : e(w) dx + ∂B R w · (Ce(u Σ0 )ν) dH 1 , and using again (7.13), we deduce that w is a solution of the minimization problem (7.2) for a.e. R > 0 with Γ ⊂ B R . Finally, an integration by parts ensures that the value of F (Γ) is independent of R > 0 and a fortiori holds for every R > 0.

Appendix. A short review of Kondrat'ev theory
We follow the notations and statements of the book [25, Sec. 6.1] that we briefly recall here in the case of the bilaplacian in the cracked plane R 2 Σ 0 . Let us consider weak solutions of the problem in weighted Sobolev spaces of type V β (R 2 Σ 0 ) (see the definition in Section 2.3) which is the core of Kondrat'ev's Theory. It is easily seen that ∆ 2 (associated with homogenous Dirichlet conditions) maps w ∈ V β (R 2 Σ 0 ) to f ∈ V −4 β (R 2 Σ 0 ). For 4 this fact is quite obvious from the definition, and for < 4, it follows from a standard extension argument (see [25,Th. 6.1.2]). Kondrat'ev theory ensures that this operator is actually of Fredholm type, and that it defines an isomorphism provided β ∈ R S and ∈ Z, where S is an exceptional countable set. In our special case it turns out to be contained in the set of half integers 1 2 Z, as for most elliptic operators (see [11]). Indeed, this set appears as the spectrum of the Mellin transform of the operator written in polar coordinates, with corresponding boundary conditions. In the language of [25] this will be called the Pencil operator, denoted by A(λ) and studied in [25,Chap. 5] (and defined pp. 197 in [25] in the case that we are interested in). The exact computations in the special case of the bilaplacian are quite standard, and can be found for instance in [26,Chap. 7.1] (see also [19,Sec. 7.2.1], but with different notations and conventions leading to slightly different characteristic equations). Let us recall here those computations, still using the language of [25].
All of them, except λ = 0 and λ = 2, have geometric and algebraic multiplicities equal to 2. The associated eigenfunctions are given by explicit functions that one can find in [26, Eq. (7.1.14) and (7.1.15)]. We shall only give the ones corresponding to λ = 3/2, which are the functions defined in (6.10) and (6.11). According to all the above facts, a direct application of [25, Th. 6.