CRACK PROPAGATION BY A REGULARIZATION OF THE PRINCIPLE OF LOCAL SYMMETRY

For planar mixed mode crack propagation in brittle materials many similar criteria have been proposed. In this work the Principle of Local Symmetry together with Griffith Criterion will be the governing equations for the evolution. The Stress Intensity Factors, a crucial ingredient in the theory, will be employed in a ’non-local’ (regularized) fashion. We prove existence of a Lipschitz path that satisfies the Principle of Local Symmetry (for the approximated stress intensity factors) and then existence of a BV -parametrization that satisfies Griffith Criterion (again for the approximated stress intensity factors).

1. Introduction. From the '60 several theoretical explanation have been proposed to predict the path along which a crack propagates when the body is under mixed mode loading; among them are widely used the Principle of Local Symmetry (PLS) [8], the Maximum Energy Release Rate [4] and the Maximum Circumferential (hoop) Stress [15]. All these theories are able to predict the crack angle at initiation (the kink) and the path with a good accuracy, at least within the order of the experimental errors. In general the crack paths obtained by the above criteria are very close and they may even coincide in some special cases. In this work we will take into account only the Principle of Local Symmetry.
Let us briefly review some classical results on PLS: [8], [5] and [1]. After the first, usually considered the original source for this criterion, the other two articles contain interesting analytical results. In both [5] and [1] it is provided an asymptotic expansion for the crack path in a small (right neighborhood) of the kink point. The classical technical machinery of analytic function theory is employed in all these three papers. In particular we would like to underlined the simplicity and applicability of the approach followed in [5] and the extreme accuracy of the analysis developed in [1].
In the (more rigorous) mathematical literature it is not easy to find a complete result on quasi-static crack propagation under mixed mode. Among the few we mention [7] and [2]. In the first the evolution is governed by another directional criterion, based on the vectorial J-integral, together with a rate-dependent regularization of Griffith's criterion. Despite these differences the paper is worth to mention since it provides, in a more rigorous setting, an asymptotic result like that of [5] and [1]; proofs are based again on a perturbation approach: first it is studied the case of a straight crack (under pure mode I) and then a linearization of its (non-linear) mixed mode perturbation. It is proved an existence result in C 1,α for α < 1/2 by means of a fixed point theorem. In [2] the spirit is different. The subject of the paper is indeed the energy release rate and more specifically, the kink, i.e. the deflection angle at initiation; however, the paper is interesting also in the perspective of studying the evolution, in particular because it highlights some issues, related to the energy release rate, which seem to appear naturally when following a minimizing movement approach.
In our paper the goal is to provide an existence result for the crack path at least is a small, but finite, right neighborhood of the kink. To pursue this task we will neither linearize nor use a perturbation of the straight case; we will face instead the non-linear problem (with a regularization). Technically we will employ a fixed point theorem and functions of bounded variations (since the evolution is quasistatic). On the other hand, our approach requires a very rigorous analysis of the regularity of solutions, and in particular of the stress intensity factors; at the actual stage, the existence of the stress intensity factors has been proved completely only in the case of straight cracks [10] and it is reasonable to expect (see [12]) that the result holds true for crack path of class C 1,1 . This regularity seems prohibitive for a direct existence result. For this reason we believe that it will be necessary to follow a much longer way: we consider first a regularized problem, for which we will prove existence and regularity of solutions, then we will use these regularized solutions to approximate the real solution. The first stage along this research direction is indeed the subject of this paper.
Let us introduce the equations that governs the evolution, i.e. the Principle of Local Symmetry and Griffith criterion. For the moment, we will not consider the above regularity issues. The evolution will be represented by a curve γ, so that γ(t) is the position of the tip. Denote by Γ t = γ([0, t]) the fracture set and by K i (t, Γ t ), for i = I, II, the stress intensity factors. Then, γ should solve K II (t, Γ t ) = 0 (Principle of Local Symmetry) , For the reasons explained above we will actually employ non-local stress intensity factors, of the form where u t is the displacement in the equilibrium configuration (at time t) ,ů t its average in a small ball, and k i (θ t ) a suitable convolution kernel, supported in the ball B r (γ(t)), and depending (in polar coordinates) on the argument ϑ t of the vector x − γ(t). The convolution kernels have been defined in order to provide a good approximation of the stress intensity factors and of their right limits by means of an approximated transfer matrix, proposed in [17] and employed in [5].
Formally we will solve first for the crack path and then for its parametrization. In particular we will find a path of class C 1 and a parametrization of class BV .
It is expected that the regularity of the path can be improved, ideally up to C 1,1 . However, the regularity of the path is intimately connected to the regularity of the displacement field at equilibrium. Resultsà la Grisvard [10] are probably too strong to hold true for C 1 cracks. A feasible alternative could be higher integrability propertiesà la Meyers [13]; to our best knowledge results of this type are not known for domains with cracks and will be developed in the nearest future. Note that an higher regularity of the path would then lead to an higher regularity of (the absolutely continuous part of) the parametrization, however it would not be possible to avoid its discontinuities: mathematically they are indeed a typical feature of rate-independent system, physically they represent non-equilibrium regimes of propagation, which are known to occur in brittle fracture. The formal statement of the existence result is contained in § 4.

2.
Setting and preliminaries. The reference un-cracked configuration is given by an open, bounded Lipschitz set Ω contained in R 2 . For technical reasons and having in mind some standard experimental settings (such as the Single Edge Notch Tension) we assume that the initial crack Γ 0 ⊂Ω is a closed line segment with one endpoint on the boundary ∂Ω. For convenience we will fix a system of coordinates with the origin in the other endpoint (the crack tip) andê 1 aligned with Γ 0 (see Figure 1). We will also assume that Ω\Γ 0 is connected and that it can be represented by the union of a finite number of Lipschitz subsets, so that Korn type inequalities hold true. Note that Ω \ Γ 0 is no longer a Lipschitz set.
In the time interval [0, T ] our physical system is described (for each t ∈ [0, T ]) by two kinematic variables: the crack set and the displacement, both depending on time. The first step is therefore to define the set of admissible cracks and the set of admissible displacements.
Considering the system of coordinates introduced above, the crack path will be represented by the graph of a Lipschitz function y belonging to the convex set (1) the parameters S, C Y > 0 will be chosen later, respectively small and big enough. In this way, denoting Γ s = Γ 0 ∪ {(x, y(x)) : x ∈ [0, s]} the domain Ω \ Γ s is still connected and represented by a finite union of Lipschitz sets. For notational convenience, we introduce also the curve γ given by γ(s) = (s, y(s)).
For the displacement field u we set a Dirichlet boundary condition on a subset ∂ D Ω of ∂Ω \ Γ 0 with H 1 (∂ D Ω) > 0. For technical reasons the boundary condition will be of proportional type, i.e. of the form u = cg, for g ∈ H 1/2 (∂ D Ω, R 2 ) and c ∈ W 1,1 (0, T ) with c(0) = 0. A proportional boundary condition, besides being realistic in many experiments, is theoretically very convenient combined with linearized elastic energy, indeed it allows to consider a single set of displacements and a time depending energy, instead of a time depending set of displacements. Therefore, given a path Γ s the space of admissible displacements will be For the boundary condition u = cg it will be sufficient to consider displacements fields of the form cu for u ∈ U (Ω \ Γ s ). On the rest of the boundary ∂(Ω \ Γ s ) \ ∂ D Ω we set an homogeneous Neumann condition. Note that ∂(Ω \ Γ s ) \ ∂ D Ω contains 4 MATTEO NEGRI both ∂Ω\∂ D Ω and the crack faces Γ ± s . We employ linearized elasticity, so the energy is with density W (Du) = Du : C[Du] = µ|ε(u)| 2 + (λ/2)|tr ε(u)| 2 (µ, λ > 0 are the Lamè coefficients). In this setting, using for instance an analogous result contained in [3], it is not hard to see that a uniform Korn inequality holds true; more precisely, there exists a constant C K such that for every y ∈ Y and s ∈ [0, S] for every u ∈ U(Ω \ Γ s ).
Since we are interested in a quasi-static evolution, given Γ s it is sufficient to consider the unique equilibrium configuration of the displacement, that is {u s } = argmin{E s (u) : u ∈ U(Ω \ Γ s )}. Note that u s depends on the path Γ s and not only on s, as the notation would suggest. For the boundary condition u = cg the equilibrium field is simply given by cu s and its energy will be c 2 E s (u s ).

Continuity of displacements .
Our proof of the existence of a crack path is based on Schauder Fixed Point Theorem. To this end it will be necessary to have at our disposal the continuity of displacements with respect to incremental and configurational variations of the crack path. The results are stated respectively in the next two Lemmas. Since we are not interested in quantitative estimates the proofs will be based only on continuity arguments; for both of them the framework will be that of Γ-convergence [6]. Proof. First of all we prove that the family {u s } of minimizers is (sequentially) pre-compact in L 2 (Ω, R 2 ). Remember that, given y ∈ Y, the sets U(Ω \ Γ s ) are increasing with respect to s. Thus, we can consider each u s to be an element of the biggest set of admissible displacements, i.e. U(Ω \ Γ S ). In the same spirit, we consider the functionals E s to be defined in U(Ω \ Γ S ), setting otherwise.
Note that E S (u s ) = E s (u s ) and that E s (u s ) ≤ E 0 (u 0 ), hence the energies E S (u s ) are uniformly bounded. Then, thanks to Korn and Poincaré inequalities, the family . As a consequence, if s n → s there exist a subsequence (not relabeled) and a limitū such that u sn →ū in L 2 (Ω \ Γ S , R 2 ). Clearly the convergence in L 2 (Ω \ Γ S , R 2 ) implies the convergence in L 2 (Ω, R 2 ). It remains to show thatū = u s . To this end, we show that the sequence E sn converges to E s (for s n → s) in the sense of Γ-convergence, with respect to the strong topology of L 2 (Ω, R 2 ). Then, by a standard result on Γ-convergence [6] it will follow thatū = u s (the unique minimizer of E s ).
The Γ-liminf inequality is a consequence of the lower semi-continuity of Next, we prove the Γ-limsup inequality. We will use a convenient density argument. Letg be a lifting of the boundary datum g withg = 0 in a neighborhood U of Γ S \ Γ 0 . Given u ∈ U(Ω \ Γ s ) we introduce the auxiliary filed v = u −g. Consider the sequence v k obtained by truncation of v and defined for a.e. x ∈ Ω by By a standard argument on Γ-convergence, it is then sufficient to find a recovery sequence for the fields u k , i.e. a sequence u sn ∈ U(Ω \ Γ sn ) such that u sn → u k in L 2 (Ω, R 2 ) and E sn (u sn ) → E s (u k ). We will actually find a sequence such that u sn → u k strongly in H 1 (Ω\Γ S , R 2 ). Unfortunately, the constant recovery sequence (u sn ≡ u k ) is not a good choice because in general u k could be discontinuous on Γ s \ Γ sn and thus it may not belong to U(Ω \ Γ sn ). This problem can be by-passed smoothing u k on Γ s \ Γ sn ; to this end we will employ a sequence of capacitary , the limit of first term vanishes by dominated convergence, the second by the definition of ψ sn . Hence, u sn → u k strongly in H 1 (Ω \ Γ S , R 2 ).

Lemma 2.2.
Let y n ∈ Y such that y n → y uniformly in [0, S]. (Denote by Γ n s the corresponding sequence of crack paths). Let u n s ∈ U(Ω \ Γ n s ) be the minimizer of the energy Then Ψ n (Γ s ) = Γ n s for every s ∈ [0, S]. Let us check that Ψ n is a bi-Lipschitz map of Ω in itself. We have For convenience, let us write DΨ n (x, y) = I + M n (x, y). Then At this point we can perform a change of variable in the energies, writing With the above change of variable the minimizer u n s of E n s is mapped toū n s = u n s • Ψ s , minimizer ofĒ n s . We check that the energiesĒ n s Γconverge to E s and that the sequenceū n s is pre-compact in the weak topology of . As a consequence, by lower semi-continuity we have the Γ-liminf inequality For the Γ-limsup inequality it is sufficient to employ the constant recovery sequence. Now we prove that the sequenceū n s is bounded in , therefore (thanks to Korn and Poincaré inequalities) u n s is bounded in H 1 (Ω \ Γ s , R 2 ). Hence the sequenceū n s is weakly pre-compact. From the standard theory of Γ-convergence we get that the minimizersū n s converge to the minimizer u s strongly in L 2 (Ω, R 2 ). To conclude the proof it is then sufficient to apply the inverse change of variable.

Stress intensity factors.
If Γ s is locally (in a neighborhood of the crack tip) a line segment then from [10] there exists a unique couple of real values K I , K II (the stress intensity factors) such that, in a small neighborhood B of the tip, the displacement u s can be represented in the form whereū s ∈ H 2 (B \ Γ s ) and for k = (3 − 4ν), and ν = λ/2(λ + µ) the Poisson ratio. Here ρ and θ denote the polar coordinates in the usual local system of coordinates with center at the tip γ(s) and polar axis along γ (s) (see Figure 2). Figure 1. Absolute (left) and local (right) systems of Cartesian and polar coordinates employed in this work. The local system translates with the crack.

Remark 1.
Employing the system of coordinates in Figure 1 instead of that in Figure 2 is not substantial, however the former simplifies several proofs since it depends only on the position γ(s) and not on the tangent γ − (s).

Approximated stress intensity factors.
In general, for a path of class C 0,1 it is not known whether a representation like (4) holds. On the base of [12] it is reasonable to expect that it holds for crack paths of class C 1,1 but this class is too restrictive to provide an existence result, at least at the present stage. Thus, a notion of stress intensity factors for a larger class of crack paths is needed. Our philosophy, to provide such a notion, is to use a suitable volume integral representation, among which there are clearly several possible choices. The one given hereafter embeds an approximation of the transfer matrix [17] and provides a good approximation in the case of straight, curved and kinked cracks. Other integral representations, such as the interaction integrals [9] and the vectorial J-integral [7] could be feasible alternatives. Finally, it is interesting to remark that in the anti-plane setting an integral representation like ours provides the exact value of the stress intensity factor for a straight crack [11]. Let us consider the system of local polar coordinates of Figure 1. For x ∈ Ω \ Γ s let ϑ denote the argument of x − γ(s). Given r > 0 let c r = (2π) −1 r −5/2 and for i = 1, 2 let k 1 (ϑ) = c r a 1 cos(ϑ/2) + a 3 cos(3ϑ/2), a 2 sin(ϑ/2) + a 4 sin(3ϑ/2) The choice of the coefficients a i and b i will be done in the sequel (see Appendix A.1). For ϕ ∈ (−π, π) let us introduce the functions where B r denotes the ball B r (γ(s)) andů s is the average of u in the ball B r 2 (γ(s)). We remark that in the notation K i there is no explicit dependence on the radius r. Note also that K i (Γ s , ϕ) does not depend on the tangent vector γ − (s).
Let us see how the functions K i (Γ s , ϕ) provide an approximation of the stress intensity factors and of their right limits. Since y is Lipschitz continuous, for a.e. s ∈ [0, S] there exists the left tangent vector γ − (s) = (1, y − (s)). This vector defines the usual local system of polar coordinates (see Figure 2) with center at the tip γ(s) and polar axis along γ − (s). Let ϕ s = arctan(y − (s)). Then, for a.e. s ∈ [0, S] the approximated stress intensity factors will be given by (see Figure 3) For Γ 0 we have ϕ 0 = 0, thus will also write Now, let us see how (8) provides approximated right stress intensity factors. For s < S, given ϕ ∈ (−π, π) consider a crack 'extension' in direction ϕ, i.e. a functioñ y ∈ Y ∩ C 1 ([s, S]) withỹ(τ ) = y(τ ) for τ ≤ s andỹ + (s) = tan ϕ.
We denote by Γ s the associated path. Then we define 1 . The next Lemma provides a representation of the right stress intensity factors K * i (Γ s , ϕ) in terms of K i (Γ s , ϕ). Lemma 3.1. With the above notation Proof. Let ϕ τ = arctan(y (τ )) for τ > s and let ϕ s = arctan(y + (s)). Denote by ϑ τ the argument of x − γ(τ ) in Ω \ Γ τ . We have to show that By Lemma 2.1 we already know that u τ → u s strongly in L 2 (Ω, R 2 ). Denote by χ τ the characteristic function of ball B r (γ(τ )). Then χ τ → χ s strongly L 2 (Ω). It follows thatů τ →ů s . Asỹ ∈ C 1 ([s, S]) we have ϕ τ → ϕ s and ϑ τ → ϑ s a.e. in Ω. As the kernel k i is continuous with respect to its argument, we get . As a consequence we get (10).

Lemma 3.2.
Given y ∈ Y the functions K i and ∂ ϕ K i are continuous with respect to the variables s and ϕ.
Proof. We will denote by ϑ s the argument of x − γ(s) in Ω \ Γ s . Remember that By Lemma 2.1 we known that u s is continuous in L 2 (Ω, R 2 ) from which it follows thatů s is continuous. Moreover, the kernels k i are Lipschitz functions, thus Note that for s → t we have ϑ s → ϑ t a.e. in Ω, thus by dominated convergence k i (ϑ s − ϕ) is continuous in L 2 (Ω, R 2 ) with respect to both s and ϕ. It follows that where k i is the derivative of k i with respect to its argument, say ϑ in (7). Arguing as above we get that also ∂ ϕ K i is continuous with respect to s and ϕ.
Using the above argument it is easy to prove the following Corollary.

Corollary 1.
Given y ∈ Y the functions K i (Γ s , ·) converges uniformly to K i (Γ t , ·) for s → t. A similar property holds true for ∂ ϕ K i (Γ s , ·).
In the sequel we will need also the continuity of K i (Γ s , ϕ) with respect to variations of the crack path. This is proved in the next Lemma. Proof. Denote by ϑ n s the argument of x − γ n (s) in Ω \ Γ n s . Given ϕ ∈ (−π, π), we can write Re-arranging the terms in the first integral we get By Lemma 2.2 we known that u n s → u s in L 2 (Ω, R 2 ) and henceů n s →ů s . The kernel k i is bounded and supported in a ball, hence k i (ϑ n s − ϕ) is bounded (uniformly with respect to ϕ) both in L 1 (Ω, R 2 ) and L 2 (Ω, R 2 ). Moreover, the kernel k i is Lipschitz continuous with respect to its argument, thus in Ω by dominated convergence we get that ϑ n s → ϑ s in L 2 (Ω) (uniformly with respect to ϕ).

Error estimate .
This section deals with the relationship between the real stress intensity factors K i and their approximations K i . To this end, it is clearly necessary to assume that the stress intensity factors exist, i.e. that a representation like u = K IûI + K IIûII +ū holds forū ∈ H 2 (B R \ Γ). In our specific setting only u 0 satisfies this (strict) requirement. Thus, to be rigorous, the estimate of this section will apply only to u 0 .
First of all, write u 0 = K I (Γ 0 )û I +K II (Γ 0 )û II +ū 0 . Next, we introduce the matrix M (ϕ) with elements M ij (ϕ) (for i = I, II and j = 1, 2) given by Then we definē Using the (column) vectors K, K,K,K with components K I , K II etc. we can write SinceK(Γ 0 , ϕ) −K(Γ 0 , ϕ) = o(1) for r 0 (see next subsection) the matrix M plays the role of the transfer matrix T . In particular we can choose the coefficients a i and b i (see Appendix A) in such a way that M (ϕ) coincides with an approximation T (ϕ) proposed in [17] and given by Note that for ϕ = 0 we have T (0) = M (0) = I. Thus, we can write Our next goal is to provide an estimate of the error termK(Γ 0 ) −K(Γ 0 ) as the support of the convolution kernels k i vanishes.
Next, write the error term in (13) as Using the Hölder regularity ofū and the fact that |k i | ≤ Cr −5/2 we can write that for every 0 < α < 1 there exists C α such that For the second term in (15) it is sufficient to estimateū(0) −ů 0 = u(0) −ů 0 . So, In conclusion we can write that for every ε > 0 there exists C ε such that (14) holds.
Arguing as in the proof of the previous Lemma we can prove also the following Corollary which provides an estimate for (11).

Corollary 2.
For every ε > 0 there exists C ε such that (for r sufficiently small) for every ϕ ∈ (−π, π). Proof. In this proof we will write explicitly the dependence on the radius r > 0 in the stress intensity factors K i . Remember that K(Γ 0 , ϕ, r) is given by Hence the equation K II (Γ 0 , ϕ, r) = 0 reads where C ij are the elements of the matrix T given by (12), i.e.

Remark 2.
Clearly, the kink angleφ 0 given by the previous Lemma depends on the radius r > 0. However, if r 0 thenφ 0 converges to the angle ϕ 0 obtained in the first part of the proof, i.e. neglecting the error E 2 . We remark that the value ϕ 0 is usually considered a good theoretical prediction of the kink angle in accordance with the experimental data (see for instance [16]).

Functional differential equation for the crack path.
Before proceeding, we remind that Γ 0 is straight, therefore there exists the stress intensity factors K i (Γ 0 ). We will assume that both K i (Γ 0 ) are positive and denote byφ 0 the angle given by Lemma 4.1. The crack path will be represented, in the absolute system of coordinates, see Figure 1, by the graph of functions in the set (1).
Given y ∈ Y and s ∈ [0, S] let V (Γ s ) = tgφ s whereφ s solves K II (Γ s ,φ s ) = 0 (we will prove existence and uniqueness ofφ s in Lemma 4.2). The crack path is found by solving the first order functional differential equation for a.e. s ∈ (0, S) First, we will prove that V is well defined and that it depends continuously on s.
Proof. By Lemma 3.2 we known that K i is continuous with respect to the variables s and ϕ.
To conclude, let us check thatφ s is continuous with respect to s. Given ε > 0 (sufficiently small) we known that Remember that K II (Γ s ,φ s ) = 0, thus, for |t − s| small enough, by uniform convergence of K II (Γ s , ·) we have It follows that the unique angleφ t such that K II (Γ t ,φ t ) = 0 must belong to the interval (φ s − ε,φ s + ε).

Fixed point problem for the crack path.
Remember that the crack path is the graph of a function in the set By Lemma 4.2, (if S is small) the "deflection angle"φ s belongs to (−arccos(1/3), 0). Remember thatφ s is the "deflection angle" in a system of coordinates translating with the crack. For this reason, we will choose C Y ≥ tan(arccos(1/3)). Let us consider on Y the functional F defined by  Proof. We will prove existence by means of Schauder Theorem. To this end we consider the set Y to be endowed with the topology of C 0 ([0, S]). The set Y is convex, closed and compact, by Ascoli-Arzelà Theorem. We have already seen that F takes values in Y. It remains to see that it is (sequentially) continuous. Let y n be a sequence in Y such that y n → y uniformly. Given s ∈ [0, S], letφ s such that By the uniform convergence of K II (Γ n s , ·) (Lemma 3.3) for n sufficiently large we have . As a consequence, the angleφ n s such that K II (Γ n s ,φ n s ) = 0 must satisfyφ s − ε < ϕ n s <φ s + ε. Thus,φ n s →φ s pointwise in [0, S] and hence V (Γ n s ) = tanφ n s → V (Γ s ) = tanφ s pointwise as well. As ϕ n s ∈ (−arccos( To conclude this section, let us consider how the approximated stress intensity factors depend on time. By linearity, denote u t,s (x) = c(t)u s (x) the equilibrium solution with boundary condition c(t)g and crack path Γ s . Since K i is linear with respect to the displacement, it turns out that the approximated stress intensity factors for u t,s are As a consequence, we can represent the evolution as the composition of the curve γ(s) = (s, y(s)) (given by Theorem 4.3) with a parametrization s(t). Indeed, de- for s(t) = 0 (i.e. for Γ t = Γ 0 ). At this point it is convenient to introduce In this way it is necessary to distinguish between Γ 0 and Γ s(t) for s(t) > 0; we will write, more simply, that K * II (t, Γ t ) = 0 for every t ∈ [0, T ].

Existence of a parametrization.
Once the path Γ s is given (by Theorem 4.3) the approximated Principle of Local Symmetry K * II (t, Γ t ) = 0 holds true for every parametrization s(t). We are therefore free to choose the parametrization in such a way that the approximated Griffith's criterion is satisfied. As we did above we define The precise statement, which characterizes the quasi-static evolution, is contained in the next Theorem.

so that discontinuities represents the non-equilibrium regimes of the evolution. Finally, if c(t) is strictly increasing the left-continuous parametrization s is unique.
Using the same argument of [14, § 6], the proof of the previous Theorem follows by the fact that K I (t, Γ s ) = c(t) K I (Γ s ) is continuous with respect to s and t.
Appendix A. Approximation of the right stress intensity factors.
A.1. Choice of the kernel. First of all, we will see how to choose the coefficients a i and b i , appearing in (7). Our choice is inspired by [17]. Writing u 0 in polar coordinates (8) takes the form Next, we introduce the matrix M (ϕ) with elements M ij (ϕ) given by It is easy to check that both the linear systems admits a unique solution if and only if 2k − 1 = 0. Remembering that k = 3 − 4ν it turns out that there exists a unique solution if and only if ν = 0.6, which is true since Poisson ratio ν is always less than 0.5. The explicit solutions are  In conclusion (thanks to the fact that u 0 = K IûI + K IIûII +ū 0 ) we get In particular, for ϕ = 0 we have K(Γ 0 , 0) = K(Γ 0 ) etc. and T (0) = I, therefore A.2. Convergence. In our analysis and in several other applications the radius of the ball B r , appearing in the evaluation of the stress intensity factors, is kept constant. As a consequence, just after initiation (for s 1) the crack set Γ s ∩ B r is not "regular" and it contains a corner point (the kink); in this case it is likely that the non-local quantities K i are not a good approximation of the stress intensity factors. More precisely, the error estimate of the previous section holds true if the representation (4) of u s is valid in a ball B R with R > r. But, if B r contains a kink then (in a small neighborhood of the kink itself) other two couples of singular functions appears; their singularity with respect to ρ (the distance from the kink) is of the type ρ π/ϕ ± 0 , where ϕ ± 0 are the inner angles at the kink. However, our choice of the convolution kernels k i has been crafted in such a way that K i provide an accurate value even close to the kink angle (this is why the coefficients a i and b i have been tailored to the transfer matrix). As we will see, this property is crucial in order to have a realistic prediction of the kink angle.
Let us try to give a more general picture. Given ϕ ∈ (−π/2, π/2) letγ(s) = s(cos ϕ, sin ϕ). In this way the (true) stress intensity factors do exist for every s and the convergence property of section 3.2 holds. Let K( Γ s ) be the vector of the (true) stress intensity factors; the corresponding vector of the approximated stress intensity factors will be K( Γ s , r) (note here the explicit dependence on the convolution radius r). Let their right limits be K(Γ 0 , ϕ) = lim s→0 + K(Γ s ) , K(Γ 0 , ϕ, r) = lim s→0 + K(Γ s , r) . The relationship between these four quantities are drawn in this diagram: