DCDIS A Supplement, Advances in Dynamical Systems, Vol. 14(S2)264 --269 Copyright@2007 Watam Press Finite Part Integrals and Hypersingular Kernels Youn-Sha Chan1, Albert C. Fannjiang2, Glaucio H. Paulino3, and Bao-Feng Feng4 1Department of Computer and Mathematical Sciences, University of Houston–Downtown One Main Street, Houston, Texas 77002, U.S.A., Email: [email protected] 2Department of Mathematics, University of California, Davis, CA 95616, U.S.A. 3Department of Civil and Environmental Engineering, University of Illinois 2209 Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, U.S.A. 4Department of Mathematics, The University of Texas–Pan American, 1201 W. University Drive, Edinburg, Texas 78541, U.S.A. y Abstract. Singular integral equation method is one of the most effective numerical methods solving a plane crack problem in frac- ture mechanics. Depending on the choice of the density function, very often a higher order of sigularity appears in the equation, and βx we need to give a proper meaning of the integration. In this article G(x) = G 0 e we address the Hadamard finite part integral and how it is used to solve the plane crack problems. Properties of the Hadamard finite part integral will be summarized and compared with other type c d x of integrals. Some numerical results for crack problems by using 2 a Hadamard finite part integral will be provided. Keywords. singular integral equations; Hadamard finite part in- tegrals; fracture mechanics; integral transform method. Figure 1: Antiplane shear problem for a nonhomogeneous AMS (MOS) subject classification: 35, 44, 45, and 73. βx material. Shear modulus G(x) = G0e ; c and d repre- sent the left and right crack tip, respectively; a is the half crack length. 1 Introduction ment vector w(x, y) is In general, the solution to the crack problems in the linear elastic fracture mechanics (LEFM) often leads to a system ∂w(x, y) 2w(x, y) + β = 0 (1) of Cauchy type singular integral equations [4, 5, 16] ∇ ∂x with the mixed boundary conditions a d φ (t) J d i i dt+ k (x, t)φ (t) dt+b φ (x) = p (x) , π − t x ij i i i i w(x, 0) = 0 , x / [c, d] , !c j=1 !c + ∈ (2) − " σyz(x, 0 ) = p(x) , x (c, d) , ∈ where c < x < d, a , b (i = 1, 2, , I) are real constants, where p(x) is the traction function along the crack sur- i i · · · and the kernels kij(x, t) are bounded in the closed domain faces (c, d). By a process of Fourier integral transform (x, t) [c, d] [c, d]. Each function p (x) is known and PDE (1) can be reduced to a hypersingular integral equa- ∈ × i given by the boundary condition(s). Functions φi(x) are tion [2]: the unknowns of the problems, also called by the density G eβx d 1 β functions which often are the derivatives of the displace- 0 = + +N(x, t) w(t, 0+) dt=p(x) , ments. However, if the unknown density function is cho- π (t x)2 2(t x) !c # − − $ sen to be the displacement, say wi(x), then the order of (3) singularity increases. Thus, a formulation of (a system where c < x < d, and N(x, t) takes an integral form of) hypersingular integral equations is made. 2 ∞ β √ξ cos[(t x)ξ] The choice of different unknown density function in the N(x, t)= − − dξ formulation leads to different order of singularity for the !0 2√ξ+ 2 A(ξ, β)+ξ ξ+ A(ξ, β) integral equation. For instance, consider a mode III crack % & () * problem in a nonhomogeneous elastic medium with the ' ' 3 2 shear modulus variation G(x) = G eβx (illustrated in ∞ β B(ξ, β) ξ sin[(t x)ξ]/2 0 − − dξ Figure 1), then the governing a partial differential equa- − & !0 B(ξ, β)'+ξ2 +√2 ξ 2ξ2 +β2 +2 B(ξ, β) tion (PDE) in terms of the z component of the displace- | | % ( &' ) ' * 264 with A(ξ, β) = ξ2 + β2 and B(ξ, β) = ξ4 + β2ξ2. Un- where x > c, and less a proper meaning of integration is given , the first d φ(t) d φ(t) integral in equations (3) is meaningless; the integral is dt := lim dt + φ(x) ln ' , regularized by “Hadamard finite part integral”, and we − t x " 0 t x !x − → +!x+" − , have used “double-bar integral” to denote it. Hadamard finite part (HFP) integral was first introduced by Jacques where x < d. Hadamard [8] to solve some linear PDE, which can be con- sidered as a generalization of the Cauchy principal value 2.2 HFP Integral (CPV) integral [6]. CPV integral does not work for a higher singularity. For instance, consider φ(t) = 1 and x = 0 in 2 HFP and CPV Integrals d φ(t) dt , c < x < d , (7) (t x)2 HFP integral is a generalization of CPV integral, thus let !c − us look at CPV integral first. that is, d dt , c < 0 < d . t2 2.1 CPV Integral !c The integral is not convergent, neither does the principal Equations that involve integrals of the type value exist, since d φ(t) dt 1 1 2 dt , x < 1 (4) = lim + 2 " 0 −c t x | | [c, d] ( ", ") t c − d ' ! − ! \ − → % ( in not integrable in the ordinary (Riemann or Lebesgue is not finite. Hadamard finite part integral is defined by integral) sense because of the kernel 1/(t x) is not in- disregarding the infinite part, 2/' , and keeping the finite tegrable over any interval that includes the− point t = x. part, i.e. d Thus, it is regularized by CPV integral [10, 13]: dt 1 1 = 2 = . (8) c t c − d d x " d ! φ(t) − φ(t) φ(t) dt := lim dt + dt , Definition 1 ( Hadamard finite part integral ) Let ' > − t x " 0 t x t x 0, and denote !c − → +!c − !x+" − , (5) where c < x < d. Notice that the '-neighborhood about F (', x) = f(t, x) dt , c < x < d , [c, d] (x ", x+") the singular point x = t must be symmetric, and it is how ! \ − CPV integral works out for canceling off the singularity. where the singularity appears at the point t = x. If F (', x) For the existence of the CPV integral, the function φ(x) is decomposed into in (5) needs to be at least H¨older continuous on (c, d), that is, φ(x) C0,α(c, d). This requirement of regularity can F (', x) = F0(', x) + F1(', x) , be easily∈checked by following manipulation: and " 0 d lim F0(', x) < , F1(', x) → , φ(t) " 0 ∞ → ∞ dt → − t x then the finite part integral is defined by keeping the “finite !c − φ(t) φ(x) dt part”, i.e. = lim − dt + φ(x) d " 0 t x " t x t x " t x → +!| − |≥ !| − |≥ , − − = f(t, x) dt = lim F0(', x) . d d φ(t) φ(x) dt c " 0 = − dt + φ(x) . (6) ! → c t x −c t x Notice that HFP integral can be considered as a gen- ! − ! − eralization of the CPV integral in the sense that if the Thus, for any φ C0,α, α > 0, the first integral on the ∈ principal value integral exists, then they give the same right side of (6) is an ordinary Riemann integral and the result [6]. We shall define the HFP integral for integrals second integral is with quadratic singularity as in (7). Denote by Cm,α(c, d) d the space of functions whose m-th derivatives are H¨older dt d x continuous on (c, d) with index 0 < α 1. = log − , c < x < d. ≤ −c t x x c ! − − Definition 2 If φ(x) C1,α(c, d), then ∈ Although Cauchy principal value integral is defined for d φ(t) an interior point in (c, d) above, it can be evaluated sep- = dt := (t x)2 arately on both sides of the end points: !c − x " d x x " − φ(t) φ(t) 2φ(x) φ(t) − φ(t) lim 2 dt + 2 dt .(9) dt := lim dt φ(x) ln ' , " 0 / c (t x) x+" (t x) − ' 0 − t x " 0 t x − → ! − ! − !c − → -!c − . 265 The condition φ(x) C1,α(c, d) is required for the exis- For φ Cn,α(c, d) L1+, the first integral on the right tence of the defined ∈HFP integral [12]. side of∈(13) is an ordinary∩ Riemann integral. Also, with Following observation may help to understand Defini- (13) in hand, integration by-parts formula holds for finite tion 2 for HFP. By a step of integration by-parts, the first part integrals [15]. integral under the limit ' 0 in (9) can be written as → Proposition 3 For φ Cn,α(c, d) L1+ x " x " ∈ ∩ − φ(t) φ(x ') c − φ%(t) d d φ%(t) φ(t) 2 dt = − + dt . c (t x) ' − c x c t x) = n dt = n = n+1 dt ! − − ! − c (t x) c (t x) ! − ! − Similarly, φ(d) φ(c) + n n , n 1 d d (d x) − (c x) ≥ φ(t) φ(x + ') d φ%(t) − − dt = + dt . (t x)2 ' − d x t x) and for φ Cα(c, d) L1+ !x+" − − !x+" − ∈ ∩ Thus, the term 2φ(x)/' in (9) will kill the singularity d − φ%(t) log t x dt [φ(x ') + φ(x + ')]/', and under the assumption that c | − | − 1,α ! φ(x) C (c, d) Definition 2 indeed takes the finite part d φ(t) of the∈integral according to Definition 1.