PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO

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1 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE V.A. KOVTUNENKO Overview: 1. Motivation 1.1. Evolutionary problem of crack propagation 1.2. Stationary problem of crack equilibrium 1.3. Interaction (contact+cohesion) between crack surfaces 2. Stationary cohesion problem for crack 2.1. Variational formulation of cohesion problem 2.2. Necessary optimality condition as a hemivariational inequality 2.3. Sufficient optimality condition as a saddle-point problem 2.4. Primal-dual formulation 2.5. Boundary-value formulation 3. Evolution of crack via optimization problem 3.1. Properties of the energy function with respect to parameter of crack length 3.2. Existence of a non-singular solution Introduction For the evolutionary problem describing crack propagation in a solid with allowance for the irreversible work due to cohesion contact between the crack opposite surfaces, general variational principles are proposed and investigated. The resulting crack model implies a quasibrittle fracture phenomenon. From an optimization point of view, the model is described by the minimization of a nonconvex and nondifferentiable functional of the total potential energy subject to Institute for Mathematics, Karl-Franzens-University of Graz, Heinrichstr. 36, 8010 Graz, Austria, and Lavrent ev Institute of Hydrodynamics, Novosibirsk, Russia; kovtunenko@hydro.nsc.ru. 1

2 2 V.A. KOVTUNENKO contact conditions. In this context, necessary and sufficient optimality conditions for the minimization problem do not coincide. The necessary optimality condition is represented as a hemivariational inequality. The sufficient optimality condition is expressed as a saddle point problem, which treats the displacement, contact and cohesion forces at the crack as independent state variables. For the optimal crack, data on the H 2 -smoothness of the displacement field in the solid and, hence, on the finiteness of the stress at the crack tip, are obtained. The solvability of the variational problem (i.e., the existence of an optimal crack) is proved for a curvilinear crack propagation path specified a-priori. For the particular case of a straight path, a generalized criterion of crack growth is proposed. 1. Motivation 1.1. Evolutionary problem of crack propagation: For solid occupying domain Ω R 2 with fixed interface Σ, for load f(t) with respect to time parameter t 0 find a crack Γ l(t) of unknown length l(t) propagating along Σ. x 2 Ω f(t) Σ 0 l(t) L Γ l(t) x Stationary problem of crack equilibrium: For solid occupying domain Ω R 2 with fixed crack Γ l of length l along Σ, for load f find a displacement u l in Ω \ Γ l.

3 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE 3 x 2 Ω f Σ 0 l L Γ l x Interaction (contact+cohesion) between crack surfaces ν contact opening 2 normal displacement u ν on crack surface Γ + normal displacement u ν on crack surface Γ Possible constraint: contact between crack surfaces results in the [u ν ] := (u ν ) Γ + (u ν ) Γ 0. Cohesion is expressed by forces: the interaction force q = λ + p is the sum of a contact force λ 0 for [u ν ] = 0 and a cohesion force p 0 for [u ν ] < δ. λ p [u ν ] δ 0

4 4 V.A. KOVTUNENKO 2. Stationary cohesion problem for crack Consider linear elastic solid in bounded domain Ω R 2 under load f in Ω and clamped at Ω, which contains crack Γ with normal ν and the crack surfaces Γ ±. potential energy surface energy x 2 x1 Γ + Γ ν Ω Γ Ω f For displacement vector u and standard linear elasticity tensors (i, j = 1, 2) σ ij (u) = c ijkl ε kl (u), ε ij (u) = 0, 5(u i,j + u j,i ), energy (cost) functional of the total potential energy T (u, Γ) = Π(u, Γ) + S([u ν ], Γ) is the sum of the potential energy Π(u, Γ) = 1 σ ij (u)ε ij (u) dx 2 f i u i dx, Ω\Γ and the surface energy S([u ν ], Γ) = Ω\Γ g([u ν ]) ds. Function of distribution of the surface energy is given by Γ g([u ν ]) = 2γ δ min(δ, [u ν ]) on Γ

5 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE 5 due to Barenblatt, Dugdale, Leonov Panasyuk hypothesis, in contrast to the brittle fracture with Griffith hypothesis g([u ν ]) 2γ (γ and δ are material parameters). Function g is non-differentiable and non-convex. 2γ g([u ν ]) Nice properties of the function 0 ξ g: uniformly bounded g(ξ) 2γ, nonnegative g(ξ) 0, δ [u ν ] Lipschitz continuous g(ζ) g(ξ) 2γ δ ζ ξ, concavity property g(ζ) g(ξ) 2γ δ where the Heaviside function { 1 for ζ 0, H(ζ) := 0 for ζ < 0. H(δ ξ) (ζ ξ), 2.1. Variational formulation of cohesion problem Taking into account possible contact between crack surfaces, let us introduce the set of admissible displacements K(Ω \ Γ) = {v H(Ω \ Γ), [v ν ] 0 on Γ}, H(Ω \ Γ) = {v H 1 (Ω \ Γ) 2, v = 0 on Ω}. For given f L 2 (Ω) 2, consider variational problem: Find u K(Ω \ Γ) such that (V P ) T (u, Γ) T (v, Γ) for all v K(Ω \ Γ)

6 6 V.A. KOVTUNENKO with nondifferentiable and nonconvex cost functional T = Π + S. Based on the following properties: set K(Ω \ Γ) is convex cone, v Π is quadratic and strictly convex, v S is nonnegative and Lipschitz continuous, the following theorem holds true. Theorem. There exist a solution to problem (V P ). However, solution may be not unique, necessary and sufficient optimality conditions coincide. Hierarchy of the problems: do not regularized (ε) variational problem regularized (ε) saddle-point problem ε 0 saddle- point problem (SP P ) primal-dual problem variational problem (V P ) (P DP ) hemivariational inequality boundary-value formulation (HV I) 2.2. Necessary optimality condition as a hemivariational inequality

7 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE 7 Due to the concavity of g we get the upper limit lim sup ɛ 0 g([u ν + ɛ(v u) ν ]) g([u ν ]) ɛ using the Gâteaux differentiability of v Π, 2γ δ H(δ [u ν ])[v ν u ν ], from (V P ) we arrive at the necessary optimality condition as a hemivariational inequality: (HV I) ( ) σij (u)ε ij (v u) f i (v u) i dx Ω\Γ + 2γ δ H ( δ [u ν ] ) [v ν u ν ] ds 0 for all v K(Ω \ Γ). Γ 2.3. Sufficient optimality condition as a saddlepoint problem The sufficient optimality condition for (V P ) is characterized by the following saddle-point problem: Find (u, λ) H(Ω \ Γ) M(Γ) such that (SP P ) where Lagrangian and the dual cone L(u, µ, Γ) L(u, λ, Γ) L(v, λ, Γ) for all (v, µ) H(Ω \ Γ) M(Γ), L(v, µ, Γ) := T (v, Γ) + µ, [u ν ] Γ, M(Γ) = {µ H 1/2 00 (Γ l) : µ, ξ Γ 0 for all 0 ξ H 1/2 00 (Γ)}. To prove existence of a solution to (SP P ) requires proper regularization of the nondifferentiable function g by a continuously differentiable function g ε such that 0 g ε (ξ) g(ξ), 0 g ε(ξ) 2γ δ, g ε(ξ) = g(ξ) + O(ε).

8 8 V.A. KOVTUNENKO 2.4. Primal-dual formulation Due to the concavity property of g we get the upper limit lim sup ɛ 0 g([u ν + ɛ(v u) ν ]) g([u ν ]) ɛ using the Gâteaux differentiability of v Π, from (SP P ) we arrive at the the problem: 2γ δ H(δ [u ν ])[v ν u ν ], primal-dual formulation of (P DP ) ( ) σij (u)ε ij (v) f i v i dx + λ + 2γ δ H( δ [u ν ] ), [v ν ] = 0 Γ Ω\Γ µ λ, [u ν ] Γ 0 for all (v, µ) H(Ω \ Γ) M(Γ). Formulation (P DP ) implies existence of the second Lagrange multiplier, associated to the cohesion force at Γ: p := 2γ δ H( δ [u ν ] ) L 2 (Γ) Boundary-value formulation From (P DP ) we find the normal stress at Γ: σ ν (u) = λ + 2γ δ H( δ [u ν ] ) = λ + p = q and derive formally problem: boundary-value formulation of the σ ij,j (u) = f i (i = 1, 2) in Ω; u = 0 on Ω;

9 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE 9 and on the crack surfaces Γ ± : σ τ (u) = 0, σ ν (u) 2γ δ if [u ν ] = 0, σ ν (u) = 2γ δ if 0 < [u ν ] < δ, σ ν (u) = 0 if [u ν ] > δ. 3. Evolution of crack via optimization problem Under load f(t) in domain Ω R 2, for t 0 find crack Γ l(t) of length l(t) [0, L] propagating along fixed path Σ of length L (one end-point of crack is fixed on the boundary). Ω f(t) Σ Γ l(t) l(t) 0 L Evolution of a crack along the fixed path is described by the optimization problem: For every t 0 find l(t) [0, L] such that (OP ) T ( f(t), u l(t), Γ l(t) ) T (f(t), u l, Γ l ) for all l [0, L], where u l K(Ω\Γ l ) is a solution of the variational problem: (V P ) T ( f(t), u l, Γ l ) T (f(t), v, Γl ) for all v K(Ω \ Γ l ) Properties of the energy function with respect to parameter of crack length

10 10 V.A. KOVTUNENKO For fixed t, obtained from (V P ) energy function l T (u l, Γ l ) : [0, L] R is: monotone decreasing, uniformly bounded, continuous in [0, L), lower semicontinuous in [0, L], for the rectilinear crack path Σ, continuously differentiable in (0, L), where derivative is defined by the shape derivative T T (u l+ɛ, Γ l+ɛ ) T (u l, Γ l ) l (ul, Γ l ) := lim. ɛ 0 ɛ The derivative can be expressed with a velocity V tangential to Σ and V Ω = 0: T l (ul, Γ l ) = Ω\Γ l 2γ δ min(δ, [ul ν]) ds div (V ) div (V f i )u l i dx Γ l Ω\Γ l ( div (V c ijkl )ε kl (u l )ε ij (u l ) σ ij (u l ) ( ) ) u l i,kv k,j + u l j,kv k,i dx or, equivalently, with contour integral over B around the crack tip: T l (ul, Γ l ) = σ ij (u l ) ( 1 2 ε ) ij(u l )(n k V k ) D τ (u l i)n j ds B + 2γ δ min(δ, [ul ν ]) B Γl. ν n Γ + l Γ l l B Σ τ

11 PROBLEM OF CRACK UNDER QUASIBRITTLE FRACTURE Existence of a non-singular solution Using boundedness and lower semicontinuity of l T (u l, Γ l ) we have Theorem. For every t 0, there exist a solution to problem (OP ). Existence of the minimum and monotone decreasing of l T (u l, Γ l ) result in the following principle identity: T ( f(t), u l(t), Γ l(t) ) = T (f(t), u l, Γ l ) We derive the following consequences: for all l [l(t), L]. Theorem. The solution u l(t) of (V P ) with l(t) being the solution of (OP ) obeys u l(t) H 2 (Ω \ Γ l(t) ) 2, i.e., has no singularity at the crack tip. ν Γ + l(t) Γ l(t) plastic zone τ The crack can become smaller during the quasibrittle evolution. Solutions l(t) of (OP ) are non-unique. A sketch for fixed t: 0 T (u l, Γ l ) l(t) L l

12 12 V.A. KOVTUNENKO This may require an additional criterion to choose between solutions l(t). In contrast, under brittle fracture propagating crack can NOT become smaller due to the Griffith hypothesis. This hypothesis provides reasonable choice of the solution with possible jumps of t l(t), as sketch for fixed t: 0 T (u l, Γ l ) l (t) L l + (t) l Acknowledgment. The research is supported by the 21st Century COE-program at the Department of Mathematics of Keio University, the Austrian Science Fund (FWF) (project P18267-N12), and the Russian Foundation for Basic Research (project ). References [1] Hintermüller M., Kovtunenko V.A., Kunisch K. An optimization approach for the delamination of a composite material with nonpenetration. In: R. Glowinski and J.-P. Zolesio (eds.) Lecture Notes in Pure and Applied Mathematics 252. CRC, [2] Khludnev A.M., Kovtunenko V.A. Analysis of cracks in solids. Southampton, Boston: WIT-Press, p. [3] Khludnev A.M., Kovtunenko V.A., Tani A. Evolution of a crack with kink and non-penetration// Research Report 001. Keio University, p. [4] Kovtunenko V.A. Nonconvex problem for crack with nonpenetration// Z. angew. Math. Mech V. 85. P [5] Kovtunenko V.A. Interface cracks in composite orthotropic materials and their delamination via global shape optimization// Optim. Eng V. 7. P [6] Kovtunenko V.A., Sukhorukov I.V. Optimization formulation of the evolutionary problem of crack propagation under quasibrittle fracture// Appl. Mech. Tech. Phys V. 47. P