SOME REFERENCES ABOUT APPLICATIONS • Y.S. Chan, A.C. Fannjiang, G.H. Paulino,Integral equations with hypersingular kernels–theory and applications to fracture mechanics, International Journal of Engineering Science 41 (2003) 683–720 • R. Chapko, R. Kress, L. M¨ onch, On the numerical solution of a hypersingular integral equation for elastic scattering from a planar crack, IMA J. Numer. Anal., 20 (2000), 601–619 • J.A. Cuminato, A.D. Fitt, S. McKee, A review on linear and nonlinear Cauchy singular integral and integro-differential equations arising in mechanics, J. Int. Equ. and Appl. 19 2 (2007),163-207 • M. Hori, S. Nemat-Nasser, Asymptotic solution of a class of strongly singular integral equations, SIAM J. Appl. Math., 50 , (1990), 716–725. M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 1
• M. Hori, S. Nemat-Nasser, Toughening by partial or full bridging of cracks in ceramics and fiber reinforced composites, Mech. Mat., 6 (1987), 245–269. • N.I. Ioakimidis, Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity, Acta Mech, 45 (1982), 31–47. • P. Junghanns, G. Monegato, A. Strozzi, On the integral equation formulations of some 2D contact problems, Journal of Computational and Applied Mathematics 234 (2010) 2808–2825 • A.I. Kalandiya, Mathematical Methods of Two-Dimensional Elasticity, Mir Publishers, Moscow, 1975. • A.C. Kaya, F. Erdorgan, On the solution of integral equations with strong M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 2
singularities, Quart. Appl. Math., 45 (1982), 31–47. • A. C. Kaya and F. Erdogan, On the solution of integral equations with strongly singular kernels, Quart. Appl. Math., XLV (1987), 105-122. • G. Monegato, A. Strozzi, On the contact reaction in a solid circular plate simply supported along an edge arc and deflected by a central transverse concentrated force. ZAMM Z. Angew. Math. Mech. 85 (2005), no. 7, 460–470 • E. P. Stephan and W. L. Wendland, A hypersingular boundary integral method for two dimensional screen and crack problems, Arch. Rational Mech. Anal., 112 (1990), 363-390. M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 3
◮ Non Linear Integral Equations of Prandtl’s Type We are interested in the numerical solution of integral equations of the form � 1 − ε v ( t ) ( t − x ) 2 dt + γ ( x, v ( x )) = f ( x ) , | x | < 1 , (1) π − 1 where 0 < ε ≤ 1 and the unknown function v satisfies the boundary conditions v ( ± 1) = 0 . In a two-dimensional crack problem v is the crack opening displacement defined by the density of the distributed dislocations k ( x ) as � x v ( x ) = − k ( t ) dt . − 1 M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 4
If we suppose that the nondimensional half crack length is equal to 1, the parameter ε in (1) corresponds to the inverse of the normalized crack length, measured in terms of a physical length parameter which is small relative to the physical crack length. The stress field at a crack tip has a square-root singularity with respect to the distance measured from the crack tip. This requires that the dislocation density k ( x ) is similarly singular, and it turns out that the Cauchy singular integral remains bounded at the crack tip. Thus, again we suppose that � 1 − x 2 . v ( x ) = ϕ ( x ) u ( x ) , ϕ ( x ) = Then, � 1 − ε ϕ ( t ) u ( t ) ( t − x ) 2 dt + γ ( x, ϕ ( x ) u ( x )) = f ( x ) , | x | < 1 . π − 1 M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 5
By physical reasons the functions f ( x ) and γ ( x, v ) are both nonnegative. This happens since f and γ represent the applied tensile tractions that pull the crack surfaces apart and the stiffness of the reinforcing fibres that resist crack opening, respectively. We are particularly interested in the class of problems for which γ ( x, v ) is a monotone function with respect to v , i.e. [ v 1 − v 2 ][ γ ( x ; v 1 ) − γ ( x ; v 2 )] ≥ 0 , | x | ≤ 1 , v 1 , v 2 ∈ R . For example, the case � γ ( x, v ) = Γ( x ) | v | sgn v, | x | ≤ 1 , v ∈ R , where Γ( x ) > 0 , | x | ≤ 1 , occurs in the analysis of a relatively long crack in unidirectionally reinforced ceramics. M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 6
◮ The Operators V and F 2 , 1 2 , − 1 → X ∗ = L 2 2 We consider equation (1) in the pair X = L − where ϕ ϕ � � ∗ 2 , − 1 2 , 1 2 , 1 L 2 := L 2 is the dual space of L 2 with the dual product ϕ ϕ ϕ ∞ � � u, p ϕ n � ϕ � v, p ϕ u ∈ X ∗ , � u, v � ϕ = n � ϕ , v ∈ X. n =0 We write equation (1) in operator form A ( u ) := εV u + F ( u ) = f , (2) M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 7
2 , 1 2 , − 1 2 2 where the Hypersingular Operator V : L − → L ϕ ϕ � 1 ( V u )( x ) = − d 1 ϕ ( t ) u ( t ) dt, | x | < 1 , dx π t − x − 1 is an isometrical isomorphism ∞ � ( n + 1) � u, p ϕ n � p ϕ V u = n . (3) n =0 For u, v ∈ L 2 ,s ϕ , we consider the inner product ∞ � (1 + n ) 2 s � u, p ϕ n �� v, p ϕ � u, v � ϕ, s = n � . n =0 M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 8
We recall that � � � � u ∈ L 2 ,s − t , v ∈ L 2 ,s + t � � u, v � ϕ,s � ≤ || u || ϕ,s − t || v || ϕ,s + t , . ϕ ϕ 2 ,s + 1 2 ,s − 1 2 2 Moreover, for the operator V : L − → L defined by (3), we have ϕ ϕ u ∈ L 2 ,s + 1 � V u, u � ϕ,s = || u || 2 2 , . 2 ϕ,s + 1 ϕ M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 9
Now we focalize our attention on the operator F : X → X ∗ defined by � � F ( u ) ( x ) = γ ( x, ϕ ( x ) u ( x )) . With respect to the function γ : [ − 1 , 1] × R → R we can make different assumptions, for example (A) ( v 1 − v 2 )[ γ ( x, v 1 ) − γ ( x, v 2 )] ≥ 0 , x ∈ [ − 1 , 1] , v 1 , v 2 ∈ R , and (B) | γ ( x, v 1 ) − γ ( x, v 2 ) | ≤ λ ( x ) | v 1 − v 2 | α , x ∈ [ − 1 , 1] , v 1 , v 2 ∈ R , for M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 10
some 0 < α ≤ 1 , where � 1 2 1+ α 1 − α dx [ λ ( x )] 1 − α [ ϕ ( x )] : 0 < α < 1 c α := < ∞ − 1 sup { λ ( x ) ϕ ( x ) : − 1 ≤ x ≤ 1 } : α = 1 and γ ( ., 0) ∈ L 2 ϕ . In any case we assume that t → γ ( x, t ) is continuous on R for almost all x ∈ [ − 1 , 1] and that x → γ ( x, t ) is measurable for all t ∈ R . M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 11
◮ The Solvability of the Hypersingular Integral Equation In order to show the solvability of (2) we need some Definitions and some Lemmas. An operator A : X → X ∗ is called − hemicontinuous if the function s → � A ( u + sv ) , w � is continuous on [0 , 1] for any fixed u, v, w ∈ X ; − strictly monotone if � A ( u ) − A ( v ) , u − v � > 0 for all u, v ∈ X with u � = v ; − strongly monotone if there exists a constant m > 0 such that � A ( u ) − A ( v ) , u − v � ≥ m || u − v || 2 X for all u, v ∈ X ; − coercive if there exists a function ρ : [0 , ∞ ) → R satisfying lim s →∞ ρ ( s ) = ∞ and � A ( u ) , u � ≥ ρ ( || u || X ) || u || X for all u ∈ X . M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 12
If (A) is fulfilled and if F maps X into X ∗ , then the operator Lemma 1. → X ∗ in (2) is strongly monotone with m = ε for each ε > 0 . A : X − If (B) is fulfilled, then the operator F maps L 2 ϕ into L 2 Lemma 2. ϕ , where F : L 2 → L 2 ϕ − ϕ is H¨ older continuous with exponent α . Lemma 3. [Zeidler] If the assumptions (A) and (B) are fulfilled, then the operator A is also coercive and equation (2) has a unique solution in X for each f ∈ X ∗ and ε > 0 . Theorem 1. [C.,Criscuolo,Junghanns] Let the assumptions (A) and (B) be fulfilled. Moreover, let f ∈ L 2 ,s for some s > 0 . If u ∈ L 2 , 1 implies ϕ ϕ ϕ , then the unique solution u ∗ ∈ X of equation (2) belongs to F ( u ) ∈ L 2 ,s L 2 ,s +1 . ϕ M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 13
◮ A Collocation Method Denote by kπ x ϕ nk = cos n + 1 , k = 1 , . . . , n , √ the zeros of the n -th orthonormal polynomial p ϕ n = 2 U n . Let X n denote the space of all algebraic polynomials of degree less than n and let L ϕ n be the Lagrange interpolation operator onto X n with respect to the nodes x ϕ nk , k = 1 , . . . , n . We recall that L ϕ n is defined by n � � x − x ϕ f ( x ϕ nk ) ℓ ϕ ℓ ϕ L ϕ nr n ( f ; x ) = nk ( x ) , nk ( x ) = . x ϕ nk − x ϕ nr k =1 n r =1 ,r � = k Again, we look for an approximate solution u n ∈ X n to the solution of M.R. Capobianco, Summer School on Applied Analysis 2013, Chemnitz, Germany 23-27/09/2013 14
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