On the topological derivative due to kink of a crack with non-penetration. A. M. Khludnev V. A. Kovtunenko A. Tani

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1 Research Report KSTS/RR-9/1 On the topological derivative due to kink of a crack with non-penetration by A. M. Khludnev V. A. Kovtunenko A. Tani A. M. Khludnev Lavrent ev Institute of Hydrodynamics V. A. Kovtunenko Karl-Franzens-University of Graz A. Tani Keio University Department of Mathematics Faculty of Science and Technology Keio University c 9 KSTS Hiyoshi, Kohoku-ku, Yokohama, 3-85 Japan

2 ON THE TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK WITH NON-PENETRATION A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Abstract. We define a topological derivative caused by kinking of a crack, thus, representing the topology change. Using variational methods, the objective function of the potential energy is expanded with respect to an incipient crack branch. For the sensitivity analysis we provide a Saint-Venant principle, and we decompose the solution of a model problem in the Fourier series. Introduction The problem of kinking and determining of the direction in which a crack will propagate is the subject for discussion in the literature on fracture mechanics, see [], [4], [8], [4], [6]. There is no explicit formulas of the kink angle even in the simple, linear setting of crack problems. We investigate a non-linear model problem for the crack subject to non-penetration conditions [13]. To emphasize the main difficulties arising here, in the paper we rely on a model scalar-valued problem and on a piecewise-linear path of the crack. We are aimed to derive an expansion of the objective function of the potential energy with respect to an incipient crack branch. Our task lies within the general framework of structure optimization. To account the common approaches adopted in shape and topology optimization, we refer to [1], [7], [11], [19], [5]. In spite of the known techniques, singular character of solutions dealing with cracked geometries requires always separate investigation. In the crack context, variational methods of the shape sensitivity analysis were developed in 1991 Mathematics Subject Classification. 49J4, 49K1, 49Q1, 74R1. Key words and phrases. Kink of crack, non-penetration condition, variational inequality, sensitivity analysis, shape and topology optimization. Lavrent ev Institute of Hydrodynamics, 639 Novosibirsk, Russia; khlud@hydro.nsc.ru. Institute for Mathematics and Scientific Computing, Karl-Franzens-University of Graz, 81 Graz, Austria, and Lavrent ev Institute of Hydrodynamics, 639 Novosibirsk, Russia; kovtunenko@hydro.nsc.ru. Department of Mathematics, Keio University, Yokohama 3-85, Japan; e- mail: tani@math.keio.ac.jp. 1

3 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI [13], [16], [17], [], [1], [], and other works by the authors. For the appropriate numerical methods, see [1], [8]. Utilizing the optimization approach due to [6], evolution of a crack with kink and non-penetration is described recently in [14]. The evolution process implies that it is global in time. Nevertheless, local characteristics at the time, when a kink occurs, are of especial interest. Indeed, the kinking phenomenon implies arrest of the tangential movement along the pre-kinked crack and appearance of a different branch at the point of kink. In this sense, it is close to the phenomena of crack splitting into few branches as well as appearance of a crack-like defect in a continuum. From a geometric viewpoint, these features present the change of topology. Treating changes of topology is the key difficulty in the structure analysis and optimization. The generic change of topology due to creating infinitesimal holes in a continuum was introduced successfully in the works [9], [7]. The mathematical formalism exploits a respective topological derivative of the objective function when the hole diminishes. The topological sensitivity based approach was tested numerically for the problem of crack identifiability in [3]. However, the disadvantage concerns the fact that the topological derivative was defined for a-priori smooth geometries only. Pre-described cracks are not the case. By these reasons, in the paper we adapt the notation of the topological derivative specifically for the phenomenon of kink. Denoting by r and φ the length of a branch and its angle with the pre-kinked path of a crack, respectively, we will consider the objective function (of the potential energy) r Π(r, φ) : (, R) R for arbitrary fixed φ (, π). Under suitable regularity assumptions, it can be decomposed as (A) Π(r, φ) = Π() + r Π (t, φ) dt, where Π() := Π(, φ) does not depend on φ, and the shape derivatives { 1 ( ) } (B) Π (r, φ) := lim Π(r + s, φ) Π(r, φ) s s are well defined for r > by the smooth velocity arguments of [14]. Restating a Saint-Venant principle (see [5], [18]) for the constrained crack problem, we obtain the uniform estimate (C) r Π (t, φ) dt = O(r).

4 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 3 In the general case, (C) admits bounded oscillations when r. For particular cases it can be specified in more details. If an expansion holds (D) r Π (t, φ) dt = rπ (, φ) + o(r), then we can define a topological derivative (for φ ) as the first asymptotic term in (D). Obviously, from (A) and (D) it follows that { 1 ( ) } (E) Π (, φ) = lim Π(r, φ) Π(, φ). r r Note that according to (B) and (E), generally speaking, Π (, φ) lim r Π (r, φ) in view of interchanging the limits. Using a Fourier series arguments of [1], [15], [3], for the linearized crack problem we will specify these expressions in the terms of stress intensity factors, which are of the first importance for engineers. 1. Formulation of the model problem with crack Let Ω be a bounded domain in R with Lipschitz boundary Ω and normal vector q at Ω. We assume that the origin O of a Cartesian coordinate system x = (x 1, x ) R is located strictly inside Ω. Denoting with B δ a ball of radius δ > centered at O, this assumes that R > exists such that B R Ω. A l x q r γ ν (r,φ) τ O φ Γ Ω x 1 Figure 1. Example configuration of the kinked crack Γ (r,φ). We consider the reference crack Γ as a segment AO of length l > posed in Ω along the x 1 -axis. Its right end-point lying at the origin O

5 4 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI will be associated with the point of kink. We specify shape parameters r [, R] and φ (, π) such that kinked cracks Γ (r,φ) will be formed by two parts: fixed one Γ and varying branches γ (r,φ). The branch γ (r,φ) is assumed to be a rectilinear segment of the length r starting from O with the kink of angle φ counter-clockwisely from the x 1 -axis. An example configuration is illustrated in Figure 1. The tangential vector τ and the normal vector ν at Γ (r,φ) are: { τ() = (1, ), ν() = (, 1) on Γ, τ(φ) = (cos φ, sin φ), ν(φ) = ( sin φ, cos φ) on γ (r,φ). With we denote Ω \ Γ (r,φ). For the further use we fix the radius < R < l of the ball B R inscribed in Ω, thus the left end-point A is located outside of B R. Starting modeling we fix r and φ. To model a solid which occupies the domain with crack, we rely on the scalar-valued setting of the problem. Let Ω consist of two parts Γ N and Γ D such that meas (Γ D ) >. The volume force f C 1 (Ω) and the boundary traction g L (Γ N ) are given. For x, unknown displacements u(x) are assumed to be zero at Γ D. Along the crack, they are restricted by non-penetration conditions due to possible contact between the opposite crack faces: (1) [u] := u Γ + u (r,φ) Γ on Γ (r,φ). (r,φ) The positive Γ + (r,φ) and the negative Γ (r,φ) faces can be distinguished geometrically as the limit of points x going to Γ (r,φ) from above and from below, respectively. The potential energy of a solid is represented by the domain-dependent functional () Π(u; ) = 1 u dx fu dx gu dx Γ N defined over the Sobolev space (3) H( ) = {u H 1 ( ) : u = on Γ D }. It is equipped with the norm (4) u H( ) = u dx, which is equivalent to the standard H 1 -norm due to the Dirichlet boundary condition at Γ D. The non-penetration condition (1) accounts

6 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 5 for the set of admissible displacements (5) K( ) = {u H( ) : [u] on Γ (r,φ) }, which is a convex cone in H( ). The equilibrium of a solid with crack is described by the following constrained minimization problem: Find u (r,φ) K( ) such that (6) Π(u (r,φ) ; ) Π(v; ) for all v K( ). Optimality conditions for (6) are expressed by the variational inequality u (r,φ) (v u (r,φ) ) dx f(v u (r,φ) ) dx (7) + Γ N g(v u (r,φ) ) dx for all v K( ). By the Lax Milgram theorem, there exists the unique solution to problem (6), equivalently, (7). Variational inequality (7) describes a weak solution to the boundary-value problem: (8a) u (r,φ) = f in, (8b) u (r,φ) = on Γ D, (8c) [ u (r,φ) ] =, ν u (r,φ) ν u (r,φ) u (r,φ) q, = g on Γ N, [u (r,φ) ], [u (r,φ) ] = on Γ (r,φ). ν To give an exact sense to the boundary terms in (8c), we introduce a Lions Magenes space H 1/ (Γ (r,φ) ) which is equipped with the norm: u = H 1/ (Γ (r,φ)) u H 1/ (Γ (r,φ) ) + u (x(s)) dist (x(s), Γ (r,φ) ) dx(s), (9) u H 1/ (Γ (r,φ) ) = u (x(s)) dx(s) + Γ (r,φ) Γ (r,φ) Γ (r,φ) Γ (r,φ) u(x(t)) u(x(s)) x(t) x(s) dx(s) dx(t) for parameters s, t ( l, r) of the length along the crack Γ (r,φ). The function dist (x(s), Γ (r,φ) ) of distance from x(s) to the crack end-points Γ (r,φ) has the order l + s as s l, and r s as s r. With

7 6 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI H 1/ (Γ (r,φ) ) we denote the formally dual space to H 1/ (Γ (r,φ) ). Then the following proposition holds, see [13] for its detailed proof. Proposition 1. The solution of (7) possesses the properties: (1) u (r,φ) K( ), u (r,φ) L ( ), [u (r,φ) ] H 1/ (Γ (r,φ) ), u (r,φ) H 1/ (Γ (r,φ) ), ν and the solution is H -smooth up to Γ ± (r,φ) apart from the kink and end points of the crack. In what follows we keep the kink angle φ fixed and pass the branch length r to zero. For r =, data of the crack problem do not depend on φ, so we exclude φ from notation for simplicity: Γ (,φ) = Γ, Ω (,φ) = Ω, u (,φ) = u. The reference solution u K(Ω ) minimizes (11) Π(u ; Ω ) Π(v; Ω ) for all v K(Ω ), or, equivalently, satisfies the variational inequality u (v u ) dx f(v u ) dx (1) Ω + Ω Γ N g(v u ) dx for all v K(Ω ). It describes a weak solution to the respective boundary-value problem: (13a) u = f in Ω, (13b) u u = on Γ D, q = g on Γ N, [ u ] u =, ν ν, u (13c) [u ], ν [u ] = on Γ. Firstly, we find the shape derivative at finite r > defined as (14) Π Π(u (r+s,φ) ; Ω (r+s,φ) ) Π(u (r,φ) ; ) (r, φ) = lim. s s With the help of (14), second we restate the topological derivative as the following limit at r = (if it exists) (15) Π (, φ) = lim r ( 1 r r ) Π (t, φ) dt.

8 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 7 It expresses the derivative of energy at the kink point in the direction of τ(φ).. Derivative of the energy functional at r > For fixed r (, R) we can apply the regular perturbation arguments. For this reason, we construct a time-independent velocity (16) V (x) = xη(x) W 1, (R ), V = on Ω, where x η : R [, 1] is a suitable cut-off function supported in Ω such that η 1 in a ball B δ of radius δ (r, R) and η outside B R. The usual solvability arguments provide existence of a unique solution (17) Φ(t, x) C 1 ([ T, T ]; W 1, (Ω)), T >, of the Cauchy problem for a nonlinear ODE d (18) Φ(t, ) = V (Φ(t, )) for t, Φ(, x) = x. dt Since (18) is an autonomous system, we obtain the identities (19) Φ( t, Φ(t, x)) = Φ(t, Φ( t, x)) = x implying that Φ( t, x) is an inverse function to Φ(t, x). In the ball B δ where η 1, the solution to (18) can be calculated analytically as () Φ(t, x) = x e t when x e t B δ. Relations (16) () argue the following proposition, see [14] for details. Proposition. There exists T > such that, for t [ T, T ], the coordinate extention y = Φ(t, x) yields a bijective mapping between the domains and Ω (re t,φ), and between sets (5) in the following sense: (1) if u K( ), then u Φ( t) K(Ω (re t,φ)); if u K(Ω (re t,φ)), then u Φ(t) K( ). Next, we rewrite the potential energy functional () in the perturbed domain Ω (re t,φ) Π(u; Ω (re t,φ)) = 1 u dx fu dx gu dx () for u H(Ω (re t,φ)) Ω (re t,φ) Ω (re t,φ) and expand it in small t. In fact, transformation y = Φ(t, x) applied to () yields (3) Π(u; Ω (re t,φ)) = Π Φ(t)(u Φ(t); ) Γ N for u H(Ω (re t,φ))

9 8 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI with the perturbed functional (4) Π Φ(t)(u; ) = 1 ( ( u) Φ 1 Φ 1 ) u x (t) x (t) ( Φ ) det x (t) dx ( f Φ(t) u Φ ) det x (t) dx gu dx for u H( ). Using the asymptotic formula due to (18) Γ N (5) Φ(t, x) = x + t V (x) + Res t, Res t W 1, (Ω) = o(t), differentiating (5) with respect to x, and substituting the result into (4) we derive the asymptotic expansion (6) Π Φ(t)(u; ) = Π(u; ) + t Π 1 V (u, u, f; ) + Res t (u), Res t (u) c(t) ( u H( ) + const ), c(t) = o(t). With Res we denote respective residuals. The first asymptotic term is associated to a quadratic form: Π 1 V (u, v, f; ) = 1 ( u div(v ) I V x V ) v dx x (7) div(v f)v dx for u, v H( ). Note, that (7) is not symmetric with respect to u and v in the latter, linear term. For the detailed derivation of (6), see, for instance, []. Proposition 3. Since (1) and (6) hold true for problem (6), the directional derivative (14) exists, and it can be expressed by formula (8) Π (r, φ) = 1 r Π1 V (u (r,φ), u (r,φ), f; ). Proof. The complete proof is given in [14], we sketch it briefly. We consider the perturbed problem (6) for u (ret,φ) K(Ω (re t,φ)) such that minimizes (9) Π(u (ret,φ) ; Ω (re t,φ)) Π(v; Ω (re t,φ)) for all v K(Ω (re t,φ)).

10 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 9 With the help of (1) and (), it follows from (9) that u (ret,φ) Φ(t) K( ) is the unique solution minimizing (3) Π Φ(t)(u (ret,φ) Φ(t); ) Π Φ(t)(v; ) for all v K( ). Substituting u as a test function into (3) results in the uniform estimate due to (6): u (ret,φ) Φ(t) H( ) c + c 1 u H( ) + O(t). Hence, a subsequence of u (ret,φ) Φ(t) exists which converges weakly to u (r,φ). By the usual arguments of monotone operators we arrive at (31) u (ret,φ) Φ(t) u (r,φ) strongly in H( ) as t. Due to (3), (6), and (3) we evaluate the increment of energy from above: (3) Π(u (ret,φ) ; Ω (re t,φ)) Π(u (r,φ) ; ) = Π Φ(t)(u (ret,φ) Φ(t); ) Π(u (r,φ) ; ) Π Φ(t)(u (r,φ) ; ) Π(u (r,φ) ; ) = t Π 1 V (u (r,φ), u (r,φ), f; ) + Res t (u (r,φ) ), Res t (u (r,φ) ) = o(t). On the other hand, (6) implies similar estimation from below: (33) Π(u (ret,φ) ; Ω (re t,φ)) Π(u (r,φ) ; ) Π Φ(t)(u (ret,φ) Φ(t); ) Π(u (ret,φ) Φ(t); ) = t Π 1 V (u (ret,φ) Φ(t), u (ret,φ) Φ(t), f; ) + Res t (u (ret,φ) Φ(t)), Res t (u (ret,φ) Φ(t)) = o(t). For re t = r + s, we have t = ln(1 + s/r) = s/r + o(s/r). Dividing (3) and (33) with s, passing s due to (31) we infer (8). The sign of the derivative follows from the general fact that r Π(r, φ) is a nonincreasing function of the crack length. Corollary 1. If f = in B δf, δ f >, the derivative in (8) can be expressed equivalently by the domain integral over Ω \ B δ only (34) Π (r, φ) = 1 r Π1 V (u (r,φ), u (r,φ), f; Ω \ B δ ), or by the contour integral over (35) Π (r, φ) = δ { 1 ( u r u(r,φ) (r,φ) ) } dx n

11 1 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI for arbitrary δ (, δ f ). The notation uses the normal vector (36) n := x x In the general case we have on. (37) u (r,φ) H (B δ \ Γ (r,φ) ) Π (r, φ) =. Proof. Let us rewrite (8) with the help of (7) explicitly as Π (r, φ) = 1 { 1 ( r u(r,φ) div(v ) I V x V ) u (r,φ) x (38) div(v f)u (r,φ) } dx. In B δ we can take η 1, thus V (x) = x, and the density of the integral in (38) is div(xf)u (r,φ) due to identity div(x) I x x x =. x Henceforth, f = in B δ implies (34). In B R \B δ the solution u (r,φ) is H -smooth according to Proposition 1. Therefore, we can differentiate the domain integral by parts and obtain due to V = in Ω \ B R : { 1 ( u(r,φ) div(v ) I V x V } ) u (r,φ) div(v f)u (r,φ) dx x Ω \B δ = ( u (r,φ) + f)(v u (r,φ) ) dx B R \B δ { 1 } + (n V ) u(r,φ) u(r,φ) n (V u(r,φ) ) dx Γ (B R \B δ ) [ 1 ] (ν V ) u(r,φ) u(r,φ) ν (V u(r,φ) ) dx. Using relations (8), the integrals over B R \ B δ as well as Γ (B R \ B δ ) are zero. At it holds V = x. As the result, from (34) we arrive at formula (35). The detailed derivation is given in []. If u (r,φ) H (B δ \Γ (r,φ) ), then the integration by parts over the whole ball B R results in Π (r, φ) =.

12 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 11 Corollary. Due to the property that y = Φ(t, x) maps Ω and K(Ω ) into themselves, it holds (39) Π 1 V (u, u, f; Ω ) =. Indeed, (39) is argued by the proof of Proposition 3 implying that = d Π Φ(t)(u Φ(t); Ω ) Π(u ; Ω ) dt Π(u ; Ω ) = lim t t = Π 1 V (u, u, f; Ω ). 3. Convergence of the solutions as r In this section, we are aimed to evaluate with respect to r the increment of solutions, which we will denote by (4) w (r,φ) := u (r,φ) u H( ). For this aim, we will employ the normal derivatives at the crack. Indeed, following Proposition 1, by the surjectivity of the trace operator at a boundary, distribution u(r,φ) H 1/ ν (Γ (r,φ) ) is defined from the Green formula by the identity u (r,φ) ν, [v ] = ( u (r,φ) v + fv) dx + gv dx Γ (r,φ) (41) for v H( ), where, Γ(r,φ) means the duality pairing between H 1/ (Γ (r,φ) ) and its dual space H 1/ (Γ (r,φ) ). Variational inequality (7) together with (41) imply that (4) u (r,φ) ν, [v u(r,φ) ] Γ (r,φ) for all v K( ). Apart from the kink and end points of the crack, where u (r,φ) is smooth, from (4) we can derive the boundary conditions (8c) pointwisely. Similarly, we can extend the normal derivative of the solution u H(Ω ) H( ) of problem (11) from Γ to the crack Γ (r,φ) as the following distribution u ν, [v ] = ( u v + fv) dx + gv dx Γ (r,φ) (43) for v H( ). Γ N Γ N

13 1 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI It fulfills the natural boundary conditions in the following sense (44) u ν, [v u ] Γ (r,φ) for v K( ) : [v ] = on γ (r,φ), where the test functions form K(Ω ). We recall that γ (r,φ) = Γ (r,φ) \ Γ. Apart from the end points of crack Γ, smooth u satisfies relations (13c) pointwisely. Subtracting (43) from (41) gets a variational formulation for the increment w (r,φ) H( ): u w (r,φ) (r,φ) v dx = u ν ν, [v ] Γ (r,φ) (45) for all v H( ). It satisfies weakly the following boundary-value problem: (46a) w (r,φ) = in, (46b) w (r,φ) = on Γ D, w (r,φ) q = on Γ N, w (r,φ) (46c) = u(r,φ) u on Γ (r,φ). ν ν ν In what follows we evaluate the solution of (45) with respect to r. To estimate the norm of w (r,φ) from (45), we observe the following facts. Substituting u K(Ω ) K( ) into (4) as a test function provides the inequality (47) u (r,φ) ν, [w(r,φ) ] Γ (r,φ). In contrast, u (r,φ) can not be substituted into (44). By this reason, we partition Γ (r,φ) with the help of a suitable cut-off function x χ r (x) : Ω [, 1] such that satisfies the following relations: (48) χ r (x) = 1 as x γ (r,φ), χ r (x) = as x Γ \ B r. Consequently, (1 χ r )[u (r,φ) ] = at γ (r,φ), and v = (1 χ r )u (r,φ) +χ r u can be taken in (44), thus providing the inequality (49) u ν, [w(r,φ) ] Γ (r,φ) u ν, χ r [w (r,φ) ] Γ (r,φ).

14 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 13 Substituting w (r,φ) into (45), from (47) and (49) we end with the estimate u w (r,φ) dx ν, χ r [w (r,φ) ] (5) Γ (r,φ) B r, where, Γ(r,φ) B r means the duality pairing between H 1/ (Γ (r,φ) B r ) and its dual space H 1/ (Γ (r,φ) B r ). Note that the right-hand side of (5) employs the cut-off function χ r supported locally in a neighborhood of the branch γ (r,φ) only. To use this feature in the further estimation we need to restate the usual result on traces in a local sense. Lemma 1. For u H( ), continuity of the trace operator yields the following estimates holding at the crack locally in B δ with δ (r, R): [u] c H 1/ (Γ (r,φ) B r u dx + c u dx, ) δ (51) if [u] H 1/ (Γ (r,φ) B r ), B δ \Γ (r,φ) B δ \Γ (r,φ) for the jump, and for the normal derivative u c u dx, ν H 1/ (Γ (r,φ) B r) (5) B δ \Γ (r,φ) [ u ] if = and u = in B δ \ Γ (r,φ). ν Moreover, a Poincaré inequality implies that 1 (53) u dx c u dx, if δ B δ \Γ (r,φ) B δ \Γ (r,φ) All constants c are independent of δ. B δ \Γ (r,φ) u dx =. Proof. We start deriving an equivalent formulation of the H 1/ -norm from (9). In fact, for δ > r we can take a closed extention Γ δ of Γ (r,φ) B r in B δ and extend [u] with zero along Γ δ. Accounting identities 1 r s = r 1 t s dt = r 1 t s dt

15 14 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI and taking the distance function dist (x(s), (Γ (r,φ) B r )) = (r s )/, it follows from (9) that [u] = H 1/ (Γ (r,φ) B r) [u(x(s))] dx(s) (54) + Γ δ Γ δ Γ δ [u(x(t)) u(x(s))] x(t) x(s) dx(s) dx(t) = [u] H 1/ ( Γ δ ). To derive (51) (53), we employ homogeneity arguments. With the uniform extension of coordinates x = δy, integrals in the right-hand side of (54) are transformed onto the image Γ 1 of Γ δ in B 1 as [u] = [u(δy(t)) u(δy(s))] dy(s) dy(t) H 1/ ( Γ δ ) y(t) y(s) + δ Γ 1 Γ 1 Γ 1 [u(δy(s))] dy(s) max(1, R) [u(δy)] H 1/ ( Γ 1 ). In B 1, the usual trace theorem provides standard estimation [u(δy)] c H 1/ ( Γ 1 ) 1 u(δy) H 1 (B 1 \ Γ 1 ) { } = c 1 u(δy) dy + u(δy) dy, B 1 \ Γ 1 B 1 \ Γ 1 with constant c 1, which is independent of δ. Thus, applying the inverse transformation y = x/δ in B 1 \ Γ 1 we obtain [u] c u(x) dx + c H 1/ ( Γ δ ) δ u(x) dx, B δ \ Γ δ B δ \ Γ δ and together with (54) we infer (51). Conversely, an extension operator exists such that u(δy) c H 1 (B 1 \ Γ 1 ) 3 [u(δy)], H 1/ ( Γ 1 ) with constant c 3 independent of δ. This implies the inequality { u(δy) dy(s) c 3 [u(δy(s))] dy(s) B 1 \ Γ 1 Γ 1 } [u(δy(t)) u(δy(s))] + dy(s) dy(t). y(t) y(s) Γ 1 Γ 1

16 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 15 After transformation y = x/δ it reads { 1 u(x) dx c 3 δ [u(x(s))] dx(s) B δ \ Γ δ + Γ δ Γ δ Γ δ [u(x(t)) u(x(s))] x(t) x(s) } dx(s) dx(t). Since [u(x(s))] = for s > r, we can evaluate 1 [u(x(s))] r s [u(x(s))] dx(s) = dx(s) δ δ r s Γ δ Γ δ Γ δ [u(x(s))] r s Γ δ dx(s) = Γ δ r Γ δ [u(x(t)) u(x(s))] x(t) x(s) [u(x(s))] t s dx(s) dt dx(s) dx(t). As a consequence, the following uniform estimate holds with c 4 = c 3 : u dx c 4 [u] for u H 1 (B H 1/ ( Γ δ ) δ \ Γ (r,φ) ). (55) B δ \ Γ δ The distribution u/ ν H 1/ (Γ (r,φ) B r ) is defined well from a generic Green formula by the relation u ν, [v ] Γ (r,φ) B r = B δ \Γ (r,φ) ( u v v u) dx for v H 1 (B δ \ (Γ (r,φ) B r )), u = on, provided that u L (B δ \ Γ (r,φ) ) and [ u/ ν ] = at Γ δ. If u =, we evaluate the norm as u := sup u 1/ ν H (Γ (r,φ) B r ) [v ] 1/ =1 ν, [v ] Γ (r,φ) B r H (Γ (r,φ) B r) ( ) 1/ = sup u v dx c 4 u dx [[v]] H 1/ ( Γδ ) =1 B δ \Γ (r,φ) due to (55), thus providing (5). B δ \Γ (r,φ)

17 16 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI If the integral of u over B δ \ Γ (r,φ) is zero, a Poincaré inequality in B 1 reads: u(δy) dy c u(δy) dy for u(δy) dy =. B 1 \ Γ 1 B 1 \ Γ 1 B 1 \ Γ 1 Consequently, with the help of y = x/δ we obtain (53). For the following use we decompose in B R \ Γ (r,φ) : (56) w (r,φ) = w (r,φ) + W (r,φ), w (r,φ) := 1 ( φ + π π φ ) w (r,φ) dθ, using polar coordinates x = ρ(cos θ, sin θ). The polar angle θ (, φ) (φ, π) starts from the x 1 -axis counter-clockwisely around the origin O. The polar radius ρ = x = x 1 + x. Firstly, integrating (56) with respect to θ gets (57) ( φ Second, we evaluate w (r,φ). + π φ ) W (r,φ) dθ =. Lemma. The function x w (r,φ) is constant for x B R \ Γ (r,φ). Proof. We take a smooth cut-off function ξ(ρ) supported in B R and substitute v = ξ into (45). In view of [ξ ] = at the crack, the righthand side turns to be zero. Due to w (r,φ) L (B R \ Γ (r,φ) ) we can integrate by parts in B R \ Γ (r,φ). Thus, accounting ξ = at B R, [ξ ] = and [ w (r,φ) / ν ] = at Γ (R,φ) we obtain that = w (r,φ) ξ dx = w (r,φ) ξ dx ( φ = + π ) R φ R = π ρ R = π ρ B R \Γ (r,φ) { ) ρ(ρ w(r,φ) (ρ w(r,φ) ρ (ρ w(r,φ) ρ ρ ) ξ dρ ) ξ dρ. R + 1 w (r,φ) } ξ(ρ) dρ dθ ρ θ [ 1 ρ w (r,φ) θ ] ξ dρ θ=±π,θ=φ

18 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 17 Since ξ is arbitrary we conclude with the identity (ρ ) w(r,φ) = for all ρ (, R). ρ ρ A general solution to this differential equation gets w (r,φ) = c 1 + c ln ρ. But the logarithmic term would contradict to the inclusion w (r,φ) H 1 (B R \ Γ (r,φ) ). Indeed, if c, we can derive from (56) that B R \Γ (r,φ) πc w (r,φ) dx R π R ( w (r,φ) ) ρ dρ dθ ρ 1 R ρ dρ + c ( π ) W (r,φ) dθ dρ = + ρ due to (57). Henceforth, c = implies the assertion of lemma. Corollary 3. If f = in B δf then U satisfies with δ f (, R) and u = ū + U in B δf \ Γ, ū := 1 π π π u dθ, U dθ =, and ū (x) is constant for x B δf \ Γ. Proof. We take a suitable cut-off function ξ(ρ) supported in B δf and substitute v = u ± ξ into (1). In view of f = in B δf and ξ = on Γ N we obtain the identity B δf \Γ u ξ dx =. Henceforth, repeating the arguments used in the proof of Lemma implies that ū is constant in B δf \ Γ. The next two results restate a Saint Venant principle for the crack problem. Lemma 3. Apart from the kink point, the following estimate holds for r < δ δ < R: (58) w (r,φ) dx δ δ w (r,φ) dx. Ω \B δ Ω \B δ

19 18 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Proof. We have w (r,φ) H 1 (Ω \ B δ ), and w (r,φ) = due to (46a). Therefore, with the help of a regular extension Γ of Γ from its left end-point A up to the external boundary Ω, a generic Green formula can be written in D := (Ω \ Γ ) \ B δ. This gets w (r,φ) v dx = w (r,φ), v (59) q for v D H1 (D), D where the normal derivative w (r,φ) / q is defined in a distributional sense at D. The boundary D is a closed curve consisted of Ω,, and two faces of Γ \ B δ. For R < l, the crack end-point A is not contained in B R \ B δ. Then w (r,φ) is smooth in (B R \ B δ ) \ Γ up to the boundary and Γ (B R \ B δ ) due to Proposition 1. Let ξ : Ω [, 1] be a suitable cut-off function supported in B R such that ξ(x) = 1 for x B δ. With the partition v = ξv + (1 ξ)v, using the local smoothness of w (r,φ) and accounting the boundary conditions in (46), from (59) we infer that w (r,φ) w (r,φ) v dx = n v dx (6) Ω \B δ Γ (B R \B δ ) w (r,φ) ν for v H(Ω \ B δ ). ξ [v ] dx w (r,φ), (1 ξ)[v ] ν Γ \B δ Substituting v = ξu (r,φ) + (1 ξ)u into (4) and v = (1 ξ)u (r,φ) + ξu into (44) gets w (r,φ), (1 ξ)[w (r,φ) ]. ν Γ (r,φ) The next pointwise relations are derived from conditions (8c) and (13c) (61) w (r,φ) ν [w (r,φ) ] on Γ (B R \ B r ). After substitution of w (r,φ) into (6), the lines between (6) and (61) result in local estimation of the norm as w (r,φ) w (r,φ) (6) dx n w(r,φ) dx. Ω \B δ

20 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 19 To evaluate the right-hand side of (6) we observe that w (r,φ) n w(r,φ) dx = π W (r,φ) = ρ π ( W (r,φ) δ dθ + ρ ( w (r,φ) + W (r,φ) ) ( w (r,φ) + W (r,φ) )δ dθ ρ π ) W (r,φ) δ dθ w (r,φ) = w (r,φ) n W (r,φ) dx due to Lemma and (57). Therefore, we can apply to W (r,φ) the Poincaré inequality along the circle (63) π and estimate n (64) δ δ u dθ 4 w (r,φ) π π { δ { δ π w(r,φ) dx ( w (r,φ) ρ ( w (r,φ) ρ π ( u) dθ for u such that u dθ = θ π w(r,φ) ρ W (r,φ) δ dθ ) + 1 4δ (W (r,φ) ) } dθ ) + 1 δ ( W (r,φ) θ ) } dθ = δ w (r,φ) dx. On the other hand, differentiating with respect to δ we find that d w (r,φ) 1 dx = lim dδ ε ε w(r,φ) dx (65) Ω \B δ = w (r,φ) dx. (B δ+ε \B δ ) Ω Combining estimates (6) (65) implies the differential inequality w (r,φ) d dx δ w (r,φ) dx. dδ Ω \B δ Ω \B δ Applying a Grönwall inequality we arrive at the desired relation (58).

21 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Lemma 4. Around the kink point, if f = in B δf, the following estimate holds for < δ δ < min(r, δ f ): u δ dx u dx. (66) δ B δ \Γ B δ \Γ Proof. The statement for a general inhomogeneous load is proven in [15]. As f =, we simplify the proof with the arguments used in Lemma 3. The extended crack Γ (R,φ) splits a ball B δ into two subdomains which we denote by B + δ and B δ. Since u H 1 (B ± δ ) and u = f L (B ± δ ), the normal derivative u / q is well defined at the boundaries B ± δ in a distributional sense. From Green formulas holding in B± δ, due to f = in B δ we have u v dx = u q, v for v H 1 (B (67) ± B ± δ ). δ B ± δ For v H 1 (B δ \ Γ ), we apply to [v ] the partition 1 = χ r + (1 χ r ) at B + δ B δ with a non-negative cut-off function χ r suitably supported in B δ and satisfying (48). Accounting the local smoothness of u in B R \ B r, using [ u / ν ] = at Γ (R,φ) and [v ] = at Γ (R,φ) \ Γ result (67) in the identity (68) u v dx = B δ \Γ Γ B δ u n v dx u ν (1 χ r)[v ] dx u Substituting v = (1 ± χ r )u into (44) gets ν, χ r [v ] Γ B r. u ν, χ r [u ] Γ B r =. With the help of boundary conditions (13c) holding pointwisely at the crack apart from its end-points, we obtain (1 χ r ) u ν [u ] = on Γ B δ. Thus, from (68) with v = u we infer equality (compare with (6)): (69) B δ \Γ u dx = u n u dx.

22 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 1 Differentiating the left-hand side of (69) with respect to δ similarly to (65) implies the relation d (7) u dx = u dx. dδ B δ \Γ In view of Corollary 3 we can repeat for u the arguments used in (64) and evaluate the right-hand side of (69) as u (71) n u dx δ u dx. From (69) (71) we conclude with the differential inequality u d dx δ u dx. dδ B δ \Γ B δ \Γ Applying the Grönwall inequality results in (66). Finally, we state the main result of this section. Theorem 1. If f = in B δf with δ f (, R), the increment of solutions w (r,φ) = u (r,φ) u converges to zero as r strongly in H(Ω (δ,φ) ) for arbitrary fixed δ (, δ f ) with the following uniform estimates: (7) w (r,φ) H(Ω(r,φ) ) c r, (73) w (r,φ) H(Ω \B δ ) c r. Proof. The proof is based on evaluation of (5) due to Lemma 1 Lemma 4. Indeed, the right-hand side of (5) admits the Cauchy inequality w (r,φ) H( ) = w (r,φ) dx u ν, χ r [w (r,φ) ] Γ (r,φ) B r u χ 1/ r [w (r,φ) ] ν H (Γ (r,φ) B r ) 1/ H (Γ (r,φ) B r). Applying estimate (5) with δ = r to u, which obeys u = f = in B r, gets further (74) w (r,φ) H( ) c 1 u H(Ω \B r ) χr [w (r,φ) ] 1/ H (Γ (r,φ) B r ) with constant c 1 which does not depend on r.

23 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI We assume that the cut-off function χ r in (74), which satisfies relations (48) in B r, can be extended appropriately to B r such that (75) χ r (x) 1, χ r (x) c r for x B r. In view of Lemma, constant w (r,φ) can be avoided from the right-hand side of (74) since [ w (r,φ) ] =. Henceforth, applying estimate (51) as δ = r to χ r W (r,φ) we infer that χr [w (r,φ) ] = H 1/ χr (Γ (r,φ) B r [W (r,φ) ] ) H 1/ (Γ (r,φ) B r ) c (χ r W (r,φ) ) c dx + χ r r W (r,φ) dx. B r \Γ (r,φ) B r \Γ (r,φ) Using (75) and Poincaré inequality (53) as δ = r due to (57) proceeds χr [w (r,φ) ] c H 1/ (Γ (r,φ) B r ) W (r,φ) dx + c 3 r B r \Γ (r,φ) W (r,φ) dx c 4 B r \Γ (r,φ) B r \Γ (r,φ) W (r,φ) dx. Adding w (r,φ) = to the right-hand side of the above inequality yields (76) χ r [w (r,φ) ] c H 1/ (Γ (r,φ) B r) 4 w (r,φ) dx. B r \Γ (r,φ) Ω From (74) and (76) we have the estimate ( (r,φ) w (r,φ) dx c 5 u dx B r \Γ (r,φ) B r \Γ (r,φ) ) 1/. w (r,φ) dx Now, with the help of Lemma 4 as δ = r, for fixed δ (r, δ f ) we infer that w (r,φ) dx c 5 c 6 r ( ( r δ w (r,φ) dx B δ \Γ (r,φ) ) 1/, u dx B r \Γ (r,φ) ) 1/ w (r,φ) dx thus (7). It implies also the strong convergence w (r,φ) in the H(Ω (δ,φ) )-norm as r.

24 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 3 To derive (73), we represent (7) as (77) w (r,φ) dx + Ω \B r B r \Γ (r,φ) and apply Lemma 3 for δ = r: 1 (78) w (r,φ) dx c 8 r Ω \B δ Ω \B r w (r,φ) dx c 7 r w (r,φ) dx. Evidently, (77) and (78) imply the estimate 1 w (r,φ) dx + w (r,φ) dx c r, r Ω \B δ B r \Γ (r,φ) thus (73). Corollary 4. A weak limit ẇ φ H(Ω \ B δ ) exists such that 1 (79) w (rn,φ) ẇ φ weakly in H(Ω \ B δ ) as r n. r n Indeed, this is a direct consequence of the uniform estimate (73). 4. Expansion of the energy functional at r = The unifrom estimation of solutions from Theorem 1 provides statements for the energy functional formulated below. During the rest of the paper we assume always that f = in B δf with fixed δ f (, R). Proposition 4. The sequence of derivatives Π (r, φ) is bounded uniformly with respect to r. Proof. Due to f = in B δf we can apply Corollary 1 and rewrite Π (r, φ) with the help of decomposition u (r,φ) = u + w (r,φ) as Π (r, φ) = 1 r Π1 V (u, u, f; Ω \ B δ ) (8) ) + Π 1 V (u, w(r,φ), f; Ω \ B δ + Π 1 V (w (r,φ), w(r,φ), ; Ω \ B δ ). r r Using Corollary and estimate (73) yields the assertion. For fixed s >, definition (14) and Proposition 3 imply expansion of Π(s + r, ) with respect to r > in the form Π(u (s+r,φ) ; Ω (s+r,φ) ) = Π(u (s,φ) ; Ω (s,φ) ) + r Π (s + t, φ) dt.

25 4 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Passing here to the limit as s due to the strong convergence (31) and Proposition 4, the Lebesgue dominated convergence theorem gets (81) Π(u (r,φ) ; ) = Π(u ; Ω ) + r Π (t, φ) dt. Details arguing passage to the limit can be found in [14]. Moreover, we employ decomposition (8) to Π (t, φ) in (81) for t (, r) and conclude with the following result. Theorem. The expansion of the potential energy holds as r : (8) and (83) Π(r, φ) = Π() + r Π 1 V r for arbitrary fixed δ (, δ f ). Π 1 V ) (u, w(t,φ), f; Ω \ B δ dt + O(r ), t ) (u, w(t,φ), f; Ω \ B δ dt = O(r), t Corollary 5. If the limit function ẇ φ H(Ω \ B δ ) from Corollary 4 is unique and it satisfies in Ω \ B δ the relation w (r,φ) = rẇ φ + Res r, Res r H 1 (Ω \B δ ) = o(r), then the topological derivative in (15) exists uniquely given by Π (, φ) = Π 1 V (u, ẇ φ, f; Ω \ B δ ). In the general case, Theorem guarantees existence only of the limit superior and the limit inferior of the quotient in (15). For particular cases, formulas in Theorem can be specified in more details. For this reason, we rely on a linearized setting of the crack problem (8) in the next section. 5. Specification of expansions for a linear problem Avoiding the non-penetration constraint (1) will result in linear formulation of the crack problem. For this particular case, we express the integral Π 1 V in expansion (8) in the terms of stress intensity factors. We restate problems (6) (8): Find u (r,φ) H( ) such that (84) Π(u (r,φ) ; ) Π(v; ) for all v H( ),

26 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 5 which is equivalent to the variational equation (85) u (r,φ) v dx = fv dx + gv dx for v H( ), Γ N and it describes a weak solution to the linear boundary-value problem: (86a) u (r,φ) = f in, (86b) u (r,φ) = on Γ D, u (r,φ) u (r,φ) q = g on Γ N, (86c) = on Γ (r,φ). ν At r =, the reference solution u H(Ω ) satisfies (compare with (11) (13)): (87) Π(u ; Ω ) Π(v; Ω ) for all v H(Ω ), dx = fv dx + gv dx for all v H(Ω ), variational equation (88) u v Ω Ω Γ N and the respective boundary-value problem: (89a) u = f in Ω, (89b) u = on Γ D, u q = g on Γ N, u (89c) ν = on Γ. All the previous results remain true for the linear problems (84) (89). In what follows we will refine expansions of the solutions (56) with the first-order asymptotic terms in a Fourier series with respect to the polar angle θ at the point of kink. Proposition 5. Around the kink point, the following expansion holds (9) u = ū + K ρ sin θ + U 1 in B δf \ Γ with the constant stress intensity factor K R given by (91) K = 1 π ρ π u (ρ, θ) sin θ dθ for arbitrary ρ (, δ f),

27 6 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI and the reminder term U 1 H 1 (B δf \ Γ ) satisfying (9a) (9b) π B δ \Γ U 1 (ρ, θ) dθ = π U 1 dx U 1 (ρ, θ) sin θ dθ = for any ρ (, δ f), ( δ ) U1 dx, < δ δ < δ f. δ B δ \Γ Proof. Indeed, consider the zero-order decomposition from Corollary 3: (93) u = ū + U in B δf \ Γ, ū := 1 π Let us define in B δf \ Γ (94) a(ρ) := 1 π π π u dθ = const, π U dθ =. U sin θ dθ, U 1 := U a(ρ) sin θ. Property (9a) follows immediately from (93) and (94). Repeating the arguments of Lemma we take a smooth cut-off function ξ(ρ) supported in B δf and substitute v = ξ sin θ/ into equation (88). Accounting f = in B δf, decomposition (94) and (9a) get ( = u θ ) ( θ ) ( θ ) ξ sin dx = a sin ξ sin dx = B δf \Γ π δ f = π δ f {ρ (a sin θ ) ρ { ( ρ a ) ρ ρ (ξ sin θ ) ρ B δf \Γ + 1 ρ (a sin θ ) θ + a } ( ξ dρ + π ρ a ) 4ρ ρ ξ ρ=δf. ρ= (ξ sin θ) } dρ dθ θ Since ξ is arbitrary we derive the ordinary differential equation ( ρ a ) + a ρ ρ 4ρ =, ρ (, δ f). Its general solution implies (95) a(ρ) = K ρ + c ρ, K, c R.

28 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 7 Using (95) we can evaluate from below the norm B δf \Γ u dx = π δ f { ( a ) a } ρ + dρ + ρ 4ρ B δf \Γ U 1 dx π δ f ) (K + c dρ = +. ρ This fact contradicts to u H 1 (B δf \ Γ ) and concludes necessarily that c =. From (95) with c =, (93) and (94) we infer (9) and (91). To derive (9b) we repeat the arguments of Lemma 4 modified for U1. Substituting expansion of the solution (9) into equality (69) and using (9a) gets for δ (, δ f ): u dx = B δ \Γ = π K δ + u n u dx = π K δ + which implies the equality (96) U1 dx = B δ \Γ B δ \Γ U 1 dx U 1 n U 1 dx, U 1 n U 1 dx. Moreover, (9a) guarantees for U 1 the following Poincaré inequality π (U 1 ) dθ π ( U 1 θ ) dθ in comparison with (63). Henceforth, from (96) we can estimate U1 δ π { ( U ) dx δ 1 1 } + ρ δ (U 1 ) (δ) dθ B δ \Γ δ U1 dx = δ d dδ B δ \Γ U 1 dx. Applying the Grönwall inequality ends with the desired property (9b). Note that U 1 obeys the H -smoothness property, see [1]. A generalization of this result for the nonlinear crack problem (1) is given in [15].

29 8 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Next we consider the increment of solutions w (r,φ) := u (r,φ) u, which describes the boundary-value problem (compare with (46)): (97a) w (r,φ) = in, (97b) w (r,φ) = on Γ D, w (r,φ) q = on Γ N, w (r,φ) w (r,φ) (97c) = on Γ, = u ν ν ν on γ (r,φ). Its weak solution w (r,φ) H( ) satisfies the variational equation (98) w (r,φ) v dx = u ν, [v ] Γ (r,φ) for all v H( ). Accounting represention (9), we state the following auxiliary result. Lemma 5. The solution of equation (98) admits the decomposition (99) w (r,φ) = K cos φ h(r,φ) + Q 1, Q 1 H(Ω(r,φ) ) = O(r), Q 1 H(Ω \B δ ) = O(r 3/ ) for δ (, δ f ), where h (r,φ) H( ) solves the problem 1 h (r,φ) v dx = ρ H γ(r,φ), [v ] for v H( ), Γ (1) (r,φ) H γ(r,φ) = 1 on γ (r,φ), H γ(r,φ) = on Γ. Proof. From (9), (98) and (1) we derive that Q 1 H( ) fulfills the equation (11) Q 1 v dx = U1 ν, [v ] Γ (r,φ) for all v H( ). Substituting v = χ r Q 1 +(1 χ r )Q 1 into (1) with the cut-off function χ r from (48) gets estimation due to U1 / ν = u / ν = at Γ Q 1 dx U 1 χ 1/ r [Q 1 ] ν. H (Γ (r,φ) B r ) 1/ H (Γ (r,φ) B r ) Since ( ρ sin θ/) =, then U1 = in B δf, hence we can apply estimate (5) in Lemma 1 to U1. Repeating arguments used in the

30 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 9 proof of Theorem 1 we continue the estimation ( Q 1 dx c 1 U1 dx B r \Γ (r,φ) B r \Γ (r,φ) ( ) 1/ c r U1 dx Q 1 dx c3 r( B δ \Γ (r,φ) B r \Γ (r,φ) ) 1/ Q 1 dx ) 1/ Q 1 dx due to (9b), thus Q 1 H(Ω(r,φ) ) = O(r). In Ω \ B δ the Green formula provides the following identity for Q 1 (similarly to (6)) Q 1 Q 1 dx = n Q 1 dx. Ω \B δ Henceforth, we can apply Lemma 3 to Q 1 and obtain Q 1 δ dx Q 1 dx, r < δ δ < δ f. δ Ω \B δ Ω \B δ Taking δ = r, the estimation finishes with Q 1 H(Ω \B δ ) = O(r 3/ ). From (1) it is interesting to observe that h (r,φ) in decomposition (99) is independent of the particular choice of the forces f and g, but it depends on the geometry of only. We proceed with an expansion of h (r,φ) in the Fourier series. Lemma 6. The following expansion holds ( h(r,φ) (1) h (r,φ) (ρ, θ) = + b (r,φ) (ρ) sin ) θ χ(ρ) + Q (ρ, θ) in, where h (r,φ) = const, χ is a cut-off function supported in B R, (13) b (r,φ) (ρ) = C (r,φ) (r,φ) 1 1/ ρ C for ρ > r, ρ 1/ stress intensity factors C (r,φ) ±1/ R are given by: (14) C (r,φ) 1/ = 1 π C (r,φ) 1/ = 1 π π π ( ) ρh (r,φ) sin θ ρ dθ, ρ ρ ( h (r,φ) ) sin θ dθ for ρ (r, R), ρ

31 3 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI and the residual term yields (15) Q H(Ω \B δ ) = O(r 3/ ) for fixed δ (r, R). Proof. Since (1) is a particular case of the variational equation (98), all the results of w (r,φ) remain valid for h (r,φ), too. By this reason, we can apply Lemma 3 and conclude with h (r,φ) = h (r,φ) + B (r,φ) in B R \ Γ (r,φ), h (r,φ) := 1 ( φ + π π φ ) h (r,φ) dθ = const, We expand further in B R \ Γ (r,φ) : (16) which implies that (17) π b (r,φ) := 1 π ( φ + π φ ( φ + π φ ) h (r,φ) sin θ dθ, B (r,φ) 1 := B (r,φ) b (r,φ) sin θ, π B (r,φ) 1 dθ = ) B (r,φ) dθ =. B (r,φ) 1 sin θ dθ = for ρ (, R). For arbitrary cut-off function ξ(ρ) supported in [r, R], the substitution of v = ξ sin θ/ as a test function into (1) gets = B R \B r R ( h (r,φ) θ ) { ξ sin dx = π ρ r (ρ b(r,φ) ρ ) + b(r,φ) 4ρ } ξ dρ. This proves the representation of b (r,φ) in the form of (13), similarly to (95). Then (13) and (16) result in the equality C (r,φ) (r,φ) 1/ ρ C 1/ 1 = 1 ρ π π h (r,φ) sin θ dθ for ρ (r, R). Differentiating it with respect to ρ yields (14). With the help of a cut-off function χ supported in B R and such that χ 1 in B δ, δ (r, R), we can define a function Q H( ) by ( h(r,φ) Q := h (r,φ) + b (r,φ) sin ) θ χ, Q = B (r,φ) 1 in B δ \ Γ (r,φ).

32 TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 31 The Green formula written in Ω \ B δ gets for v H(Ω \ B δ ): ( h (r,φ) v dx = Q + b (r,φ) χ sin θ ) v dx Ω \B δ Ω \B δ h (r,φ) { ( = n v dx = n B (r,φ) 1 + b (r,φ) sin θ )} v dx. Substituting v = Q as a test function here we obtain the equality, which is similar to (96), and estimate it in the same way Ω \B δ δ Q dx = (r,φ) B 1 dx = δ d dδ B (r,φ) 1 n B(r,φ) 1 dx Ω \B δ Q dx due to property (17). The Grönwall inequality provides ( Q δ ) dx Q dx, < δ δ < R. (18) δ Ω \B δ Ω \B δ Taking δ = r we proceed (18): Ω \B δ Ω \B r Q dx c 1 r Q dx c r c 3 r w (r,φ) dx + O(r 4 ) = O(r 3 ) h (r,φ) dx in view of Lemma 5 and Theorem 1. The latter estimate implies (15), and this ends the proof. From Lemma 5 and Lemma 6 we derive the following. Proposition 6. Apart from the kink point, the representation holds w (r,φ) (ρ, θ) = w (r,φ) + K cos φ ( C (r,φ) (r,φ) 1 ) 1/ ρ C 1/ sin θ (19) ρ + Q(ρ, θ), Q H(BR \B δ ) = O(r 3/ ) for fixed δ (r, δ f ). Finally, we state the main result of this section.

33 3 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Theorem 3. For the linear crack problems (84) (89), expansion of the potential energy at the point of kink as r reads (11) Π(r, φ) = Π() π 4 K cos φ r 1 t C(t,φ) 1/ dt + O(r3/ ), with the stress intensity factors K and C (t,φ) 1/ defined in (91) and (14), (111) r 1 t C(t,φ) 1/ dt = O(r). Proof. Let us calculate the integral Π 1 V (u, w (r,φ), f; Ω \B δ ) from Theorem explicitely by substituting here the representation of solutions (9) and (19). In this way we will specify the decomposition of energy (8) (83) for the linear problem in the form of (11) (111). We choose the cut-off function η(ρ) for the velocity V in (16) and the cut-off function χ(ρ) in representation (1) such that: η 1 in B δ, supp(η) B δf, χ 1 in B δf, supp(χ) B R for δ < δ f < R. Using (99) and (1) due to Corollary 1 and Proposition 6 gets 1 r Π1 V (u, w (r,φ), f; Ω \ B δ ) = K r cos φ ( Π1 V u, χ ( h(r,φ) + b (r,φ) sin θ ) ), f; Ω \ B δ + O( r), with b (r,φ) (ρ) given in (13). Integrating the following integral by parts in B δf \ B δ, where the solutions are smooth and χ 1, = Π 1 V ( u, χ ( h(r,φ) + b (r,φ) sin θ ) ), f; Ω \ B δ { u B δf \B δ ( div(v ) I V x ( h(r,φ) div(v f) + b (r,φ) sin )} θ dx, ) ( b (r,φ) sin θ )

34 B δf \B δ TOPOLOGICAL DERIVATIVE DUE TO KINK OF A CRACK 33 similarly to the proof of Corollary 1 we obtain ( Π 1 V u, χ ( h(r,φ) + b (r,φ) sin θ ) ), f; Ω \ B δ { ( = ( u + f) V ( b (r,φ) sin θ ) ) + ( b (r,φ) sin θ ) (V u )} dx + δ + { u ( b (r,φ) sin θ ) u ( b (r,φ) sin θ )} dx n n Γ (B δf \B δ ) = δ { u [ (b (r,φ) sin ) θ ν,1 ] + ( b (r,φ) sin θ ) } [u,1 ] dx ν { u ( b (r,φ) sin θ ) u ( b (r,φ) sin θ )} dx n n in view of (89a), (89c), and relations ( b (r,φ) sin θ ) = in Ω, ν ( b (r,φ) sin θ ) = on Γ due to (13). Applying decomposition (9), from Proposition 5 together with the orthogonality conditions (9a) it proceeds further { δ u ( b (r,φ) sin θ ) u ( b (r,φ) sin θ )} dx n n = K π = πk { ( ) ρ sin θ ( ) b (r,φ) sin θ δ ( ) ρ sin ρ ( ) b (r,φ) sin θ } θ θ ρ ρ { δ 4 ( δc (r,φ) 1/ C(r,φ) 1/ ) δ δ δ (C (r,φ) 1/ δ + C (r,φ) )} 1/ δ δ Thus, we arrive at the equality for the linear crack problem ρ=δ dθ = π KC(r,φ) 1/. (11) Π (r, φ) = 1 r Π1 V (u, w (r,φ), f; Ω \ B δ ) = π 4r K cos φ C(r,φ) 1/ + O( r). Henceforth, Theorem argues (11) and (111). We finish with few remarks on Theorem 3.

35 34 A.M. KHLUDNEV, V.A. KOVTUNENKO, AND A. TANI Corollary 6. If K = in (91), i.e., the solution u is H -smooth around the kink point, then (11) implies that K = Π (, φ) =, Π(r, φ) = Π() + O(r 3/ ). Corollary 7. If φ =, i.e., there is no kink, then it is known that the derivative exists and Π (, ) = /4 K, for example, see []. Therefore, from (11) and (111) we infer that C (r,) 1/ = r + o(r). Conclusion The asymptotic representations resulting formal analysis are also of practical meaning for engineers. In fact, the expansion of potential energy (11) is expressed via stress intensity factors K and C (r,φ) 1/. They can be calculated as a path-independent integrals by explicit formulas (91) and (14). The former constant K depends on the specific choice of data of the reference crack problem before kinking, while the latter (r, φ) C (r,φ) 1/ are universal functions depending on the geometry of the kinked domain only. These implicit quantities are to be determined from a generic problem of the crack kinking (1). Acknowledgment. The research is supported by the 1st Century COE-program at the Keio University, the Austrian Science Fund (FWF) (project P1411-N13), and the Siberian Branch of Russian Academy of Sciences (project N 9). References [1] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys. 194 (4), [] M. Amestoy and J.-B. Leblond, Crack paths in plane situations. II: Detailed form of the expansion of the stress intensity factors, Int. J. Solids Struct. 9 (199), [3] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method, Control Cybernet. 34 (5), [4] I.I. Argatov and S.A. Nazarov, Energy release caused by the kinking of a crack in a plane anisotropic solid, J. Appl. Math. Mech. 66 (), [5] C. Barbarosie and A.-M. Toader, Saint-Venant s principle and its connections to shape and topology optimization, Z. Angew. Math. Mech. 88 (8), 1, 3 3. [6] B. Bourdin, G.A. Francfort and J.-J. Marigo, The variational approach to fracture, J. Elasticity 91 (8), 1-3, [7] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection, Comput. Meth. Appl. Mech. Engrg. 188 ()

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