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 1
Department of Computer and Mathematical Sciences, University of Houston–Downtown One Main Street, Houston, Texas 77002, U.S.A., Email:
[email protected] 2
Department of Mathematics, University of California, Davis, CA 95616, U.S.A. 3
Department of Civil and Environmental Engineering, University of Illinois 2209 Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, U.S.A. 4
Department 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 fracture mechanics. Depending on the choice of the density function, very often a higher order of sigularity appears in the equation, and we need to give a proper meaning of the integration. In this article 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 of integrals. Some numerical results for crack problems by using Hadamard finite part integral will be provided. Keywords. singular integral equations; Hadamard finite part integrals; fracture mechanics; integral transform method. AMS (MOS) subject classification: 35, 44, 45, and 73.
1
Introduction
In general, the solution to the crack problems in the linear elastic fracture mechanics (LEFM) often leads to a system of Cauchy type singular integral equations [4, 5, 16]
G(x) = G 0 e
c
βx
d
Figure 1: Antiplane shear problem for a nonhomogeneous material. Shear modulus G(x) = G0 eβx ; c and d represent the left and right crack tip, respectively; a is the half crack length. ment vector w(x, y) is ∇2 w(x, y) + β
∂w(x, y) =0 ∂x
with the mixed boundary conditions ! J ! d " ai d φi (t) dt+ kij (x, t)φi (t) dt+bi φi (x) = pi (x) , − w(x, 0) = 0 , x∈ / [c, d] , π c t−x j=1 c σyz (x, 0+ ) = p(x) , x ∈ (c, d) , where c < x < d, ai , bi (i = 1, 2, · · · , I) are real constants, and the kernels kij (x, t) are bounded in the closed domain (x, t) ∈ [c, d] × [c, d]. Each function pi (x) is known and given by the boundary condition(s). Functions φi (x) are the unknowns of the problems, also called by the density functions which often are the derivatives of the displacements. However, if the unknown density function is chosen to be the displacement, say wi (x), then the order of singularity increases. Thus, a formulation of (a system of) hypersingular integral equations is made. The choice of different unknown density function in the formulation leads to different order of singularity for the integral equation. For instance, consider a mode III crack problem in a nonhomogeneous elastic medium with the shear modulus variation G(x) = G0 eβx (illustrated in Figure 1), then the governing a partial differential equation (PDE) in terms of the z component of the displace-
x
2a
(1)
(2)
where p(x) is the traction function along the crack surfaces (c, d). By a process of Fourier integral transform PDE (1) can be reduced to a hypersingular integral equation [2]: $ ! # G0 eβx d 1 β = + +N (x, t) w(t, 0+ ) dt = p(x) , 2 π 2(t−x) c (t−x) (3) where c < x < d, and N (x, t) takes an integral form √ ! ∞ −β 2 ξ cos[(t − x)ξ] () N (x, t) = % * dξ ' √ & ' 0 2 ξ + 2 A(ξ, β)+ξ ξ + A(ξ, β)
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&' ! ∞ β3 B(ξ, β)−ξ 2 sin[(t−x)ξ]/2 () − %&' * dξ ' √ 0 B(ξ, β)+ξ 2 + 2|ξ| 2ξ 2 +β 2 +2 B(ξ, β)
with A(ξ, β) = ξ 2 + β 2 and B(ξ, β) = ξ 4 + β 2 ξ 2 . Un- where x > c, and +! , less a proper meaning of integration is given , the first ! d d φ(t) φ(t) integral in equations (3) is meaningless; the integral is − dt := lim dt + φ(x) ln ' , "→0 regularized by “Hadamard finite part integral”, and we x t−x x+" t − 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 considered 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 ! d 2 HFP and CPV Integrals φ(t) dt , c < x < d, (7) (t − x)2 c HFP integral is a generalization of CPV integral, thus let us look at CPV integral first. that is, ! d dt , c < 0 < d. t2 c 2.1 CPV Integral The integral is not convergent, neither does the principal Equations that involve integrals of the type value exist, since % ( ! ! d dt 1 1 2 φ(t) = lim − + − dt , |x| < 1 (4) 2 "→0 c d ' t−x [c, d]\(−", ") t c in not integrable in the ordinary (Riemann or Lebesgue integral) sense because of the kernel 1/(t − x) is not integrable over any interval that includes the point t = x. Thus, it is regularized by CPV integral [10, 13]: +! , ! d ! d x−" φ(t) φ(t) φ(t) dt := lim dt + dt , − "→0 t−x t−x c c x+" t − x (5) where c < x < d. Notice that the '-neighborhood about the singular point x = t must be symmetric, and it is how CPV integral works out for canceling off the singularity. For the existence of the CPV integral, the function φ(x) in (5) needs to be at least H¨older continuous on (c, d), that is, φ(x) ∈ C 0,α (c, d). This requirement of regularity can be easily checked by following manipulation:
= =
! d φ(t) − dt t−x c +! lim
"→0
!
d
is not finite. Hadamard finite part integral is defined by disregarding the infinite part, 2/' , and keeping the finite part, i.e. ! d dt 1 1 = (8) = − . 2 t c d c
Definition 1 ( Hadamard finite part integral ) Let ' > 0, and denote ! F (', x) = f (t, x) dt , c < x < d , [c, d]\(x−", x+")
where the singularity appears at the point t = x. If F (', x) is decomposed into F (', x) = F0 (', x) + F1 (', x) , and
lim F0 (', x) < ∞ ,
"→0
!
φ(t) − φ(x) dt + φ(x) t−x |t−x|≥" ! d φ(t) − φ(x) dt dt + φ(x) − . t−x t−x c
|t−x|≥"
dt t−x
,
(6)
"→0
F1 (', x) → ∞ ,
then the finite part integral is defined by keeping the “finite part”, i.e. ! d = f (t, x) dt = lim F0 (', x) . c
"→0
Notice that HFP integral can be considered as a generalization of the CPV integral in the sense that if the Thus, for any φ ∈ C 0,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 C m,α (c, d) the space of functions whose m-th derivatives are H¨older ! d d−x dt continuous on (c, d) with index 0 < α ≤ 1. = log , c < x < d. − t−x x−c c Definition 2 If φ(x) ∈ C 1,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 c arately on both sides of the end points: 0 /! ! d x−" φ(t) 2φ(x) φ(t) -! x−" . ! x dt + dt − .(9) lim φ(t) φ(t) 2 "→0 (t − x)2 ' − dt := lim dt − φ(x) ln ' , x+" (t − x) c "→0 t−x t−x c c c
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The condition φ(x) ∈ C 1,α (c, d) is required for the existence of the defined HFP integral [12]. Following observation may help to understand Definition 2 for HFP. By a step of integration by-parts, the first integral under the limit ' → 0 in (9) can be written as ! x−" ! x−" % φ(t) φ(x − ') c φ (t) dt = − + dt . 2 (t − x) ' c − x t − x) c c Similarly, ! d ! d φ(t) φ(x + ') d φ% (t) dt = − + dt . 2 ' d−x x+" (t − x) x+" t − x)
For φ ∈ C n,α (c, d) ∩ L1+ , the first integral on the right side of (13) is an ordinary Riemann integral. Also, with (13) in hand, integration by-parts formula holds for finite part integrals [15]. Proposition 3 For φ ∈ C n,α (c, d) ∩ L1+ ! d ! d φ% (t) φ(t) = dt = n = dt n (t − x) (t − x)n+1 c c φ(c) φ(d) − , n≥1 + (d − x)n (c − x)n
and for φ ∈ C α (c, d) ∩ L1+ ! d Thus, the term −2φ(x)/' in (9) will kill the singularity φ% (t) log |t − x|dt [φ(x − ') + φ(x + ')]/', and under the assumption that c ! d φ(x) ∈ C 1,α (c, d) Definition 2 indeed takes the finite part φ(t) dt + φ(d) log |d − x| − φ(c) log |c − x| = − of the integral according to Definition 1. t −x c Another direction of viewing Definition 2 is by taking direct differentiation d/dx to (5) with Leibnitz’s rule, i.e. /! 0 2.3 Remark on HFP Integrals ! d ! d x−" d φ(t) d φ(t) φ(t) The most commonly used integration in mathematical − dt = lim dt + dt "→0 dx dx c t −/x t−x c x+" t − x analysis is Lebesgue integration. Not all the properties ! x−" ! d φ(t) φ(t) for Lebesgue integral can be carried onto finite part inte= lim dt + dt− 2 2 gral. For example, properties that involve inequality (e.g. "→0 (t − x) c x+" (t − x) $ monotone convergence theorem, Fatou’s lemma, bounded φ(x − ') + φ(x + ') (10) convergence theorem) may not be true for finite part in' tegral anymore. A simple demonstration is (see equation (8) and Reference [9]) Comparing (10) with (9), one can conclude ! 1 dt Proposition 1 If φ(x) ∈ C 1,α (c, d), then = = −2 ; 2 −1 t ! d ! d φ(t) d φ(t) dt (11) clearly, = dt = − 2 3! 3 ! (t − x) dx t −x c c 3 d 3 d 3 3 ≤ = |f (x)| dx = f (x) dx 3 3 Alternatively, one can define finite part integrals by 3 c 3 c equation (11) and deduce Definition 2 as property. Thus, for general n, HFP integrals can be defined recursively as is NOT true for finite part integral ! This is a very unpleasant outcome from HFP integral; however, fortufollows. nately, the most relying formula, integration by-parts, is Definition 3 ( Finite part integral ) Denote true for finite part integral. 1 1+ p L = L [c, d] .
3
p>1
Hypersingular Kernels
For any φ ∈ C (c, d) ∩ L , c < x < d, and n = For the derivation of hypersingular kernels, we use three 1, 2, 3, ... basic ingredients: ! d ! d φ(t) φ(t) 1 d • Finite part integrals. = = dt := dt (12) (t − x)n+1 n dx c (t − x)n c • Identity with # $ # $ ! d ! d n 1 dn 1 n d φ(t) φ(t) i = n . (14) = dt := − dt . dy n y − i(t − x) dx y − i(t − x) t−x t−x c c n,α
1+
By means of (6) and the (recursive) definition of finite part integrals, one can deduce [15] Proposition 2 2n−1 ! d ! c φ(t) − j=0 φ(j) (x)(t − x)j /j! φ(t) = dt = n (t − x)n c (t − x) d n−1 " φ(j) (x) ! d dt + = . (13) n−j j! (t−x) c j=0
• Plemelj formulas [3, 16]. ! d ! d φ(t) φ(t) lim dt = − dt + πiφ(x) , φ ∈ L1+ . "→0 c (t−x)+i' t−x c
The key point of identity (14) is that it allows one to switch the differentiation from d/dx to d/dy, and vice versa; HFP integral has been defined and addressed in previous section; for the sake of completeness, we shall briefly address Plemelj formulas.
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3.1
As the limit of y → 0+ is taken, ! ∞4 5 1 w(x, 0+ ) = lim+ √ α(ξ)eλ(ξ)y e−ixξ dξ y→0 2π −∞ ! ∞ 1 α(ξ)e−ixξ dξ , =√ 2π −∞
Plemelj Formulas
In general, the Cauchy principal value type of integrals ! d φ(t) − dt , t−x c
c
