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; 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 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. = dt + φ(d) log d x φ(c) log c x − t x | − | − | − | Another direction of viewing Definition 2 is by taking !c − direct differentiation d/dx to (5) with Leibnitz’s rule, i.e. 2.3 Remark on HFP Integrals d x " d d φ(t) d − φ(t) φ(t) dt = lim dt + dt The most commonly used integration in mathematical " 0 dx −c t x → dx / c t x x+" t x 0 analysis is . Not all the properties ! − x " ! d − ! − − φ(t) φ(t) for Lebesgue integral can be carried onto finite part inte- = lim dt + dt " 0 2 2 gral. For example, properties that involve inequality (e.g. → / c (t x) 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 Comparing (10) with (9), one can conclude (8) and Reference [9]) 1 Proposition 1 If φ(x) C1,α(c, d), then dt ∈ = 2 = 2 ; 1 t − d φ(t) d d φ(t) !− = dt = dt (11) clearly, (t x)2 dx − t x !c − !c − d d Alternatively, one can define finite part integrals by = f(x) dx = f(x) dx 3 c 3 ≤ c | | equation (11) and deduce Definition 2 as property. Thus, 3! 3 ! 3 3 for general n, HFP integrals can be defined recursively as is NOT true3 for finite part3 integral ! This is a very 3 3 follows. unpleasant outcome from HFP integral; however, fortu- nately, the most relying formula, integration by-parts, is Definition 3 ( Finite part integral ) Denote true for finite part integral. L1+ = Lp[c, d] . p>1 1 3 Hypersingular Kernels For any φ Cn,α(c, d) L1+, c < x < d, and n = For the derivation of hypersingular kernels, we use three 1, 2, 3, ... ∈ ∩ basic ingredients: d d φ(t) 1 d φ(t) Finite part integrals. = n+1 dt := = n dt (12) • c (t x) n dx c (t x) ! − ! − Identity with • d d n n φ(t) φ(t) n d 1 d 1 = dt := dt . i n = n . (14) t x − t x dy y i(t x) dx y i(t x) !c − !c − # − − $ # − − $ By means of (6) and the (recursive) definition of finite Plemelj formulas [3, 16]. part integrals, one can deduce [15] • d d φ(t) φ(t) 1+ Proposition 2 lim dt = dt + πiφ(x) , φ L . " 0 (t x)+i' − t x ∈ → !c − !c − n 1 (j) j d φ(t) c φ(t) − φ (x)(t x) /j! = dt = − j=0 − The key point of identity (14) is that it allows one to (t x)n (t x)n !c − !d 2 − switch the differentiation from d/dx to d/dy, and vice n 1 d − φ(j)(x) dt versa; HFP integral has been defined and addressed in + = . (13) j! (t x)n j previous section; for the sake of completeness, we shall j=0 c − " ! − briefly address Plemelj formulas.

266 3.1 Plemelj Formulas As the limit of y 0+ is taken, →

In general, the Cauchy principal value type of integrals + 1 ∞ λ(ξ)y ixξ w(x, 0 ) = lim α(ξ)e e− dξ y 0+ √2π d φ(t) → !−∞ 4 5 dt , c < x < d 1 ∞ ixξ − t x = α(ξ)e− dξ , (21) !c − √2π !−∞ is evaluated indirectly by using complex function the- that is, w(x, 0+) is the inverse of α(ξ). ory [11, 13]. Define By inverting the Fourier transform, one obtains

d φ(t) 1 ∞ Φ(z) = dt , α(ξ) = w(x, 0+)eixξdx c t z √2π ! − !−∞ 1 d with z not on the integration contour. The principal value = w(t, 0+)eitξdt , (22) √2π is then recovered by sending z to the point x on the inter- !c val (c, d), and the result is different as z x from above → where the first boundary condition in (2) and a change of and below. Say, define dummy variable (x t) have been applied. ↔ + Defining Φ (x) = lim Φ(x + i y ) , Φ−(x) = lim Φ(x i y ) , λ(ξ)y y 0 y 0 K(ξ, y) = λ(ξ)e , (23) → | | → − | | then the limits are and using the second boundary condition in (2), one reaches that for c < x < d d + φ(t) d Φ (x) = dt + iπφ(x) , (15) G(x) + ∞ i(t x)ξ − t x lim w(t, 0 ) K(ξ, y)e − dξdt c + ! − y 0 2π c → ! !−∞ d and G(x) + d = w(t, 0 )kernel(x t)dt = p(x) , (24) φ(t) π − Φ−(x) = dt iπφ(x) . (16) !c − t x − !c − ∞ i(t x)ξ 2 kernel(x t) = lim K(ξ, y)e − dξ . (25) Equations (15) and (16) are Plemelj formulas [11], some- × − y 0+ times called by the Sokhetski formulas. It is (15) that we → !−∞ Let K(ξ) K(ξ, 0+) = λ(ξ) and by a step of decomposi- will be using in the derivation of hypersingular kernels. ≡ Notice that φ(x) can be recovered from Plemelj formulas, tion K(ξ) = [K(ξ) K (ξ)] + K (ξ) , (26) i.e. ∞ ∞ + − Φ (x) Φ−(x) φ(x) = − . one obtains a closed form expressions of 2πi iβ ξ K (ξ) = ξ | | . (27) 3.2 Arise of Hypersingular Kernels ∞ −| | − 2 ξ This K (ξ) gives rise to the quadratic hypersingular (and To demonstrate how the hypersingular kernels arise, we ∞ go back to the PDE (1), and through Forier transform Cauchy singular) kernels by the following w(x, y) can be expressed as [2] + ∞ ξ y i(t x)ξ y 0 2 ξ e−| | e − dξ → − (28) | | −→ (t x)2 1 ∞ λ(ξ)y ixξ !−∞ w(x, y) = α(ξ)e e− dξ , (17) 4 5 − √2π + !−∞ ∞ ξ ξ y i(t x)ξ y 0 2 4 5 i| |e−| | e − dξ → − (29) ξ −→ t x where !−∞ # $ − [λ(ξ)]2 = ξ2 + iβξ , (18) 3.3 Higher Order Hypersingular Kernels and α(ξ) is to be determined by the boundary condi- For a more general and higher order of hypersingular ker- tions (2). To satisfy the far field boundary condition, nels, they can be derived by observing that limy w(x, y) = 0, we choose the root λ(ξ) to be the non-p→∞ositive real part: 1 ∞ n n ξ ξ y+i(t x)ξ k (t x, y) := i ξ | |e−| | − dξ n − 2π | | iξ 1 4 2 2 2 i 4 2 2 2 !−∞ − ξ + β ξ + ξ sgn(βξ) ξ + β ξ ξ , n √ − √ − 2 n d 1 2 2 = ( i) Im (y i(t x))− &' &' (19) π − dyn − − 9 # $ where the signum function sgn( ) is defined as n · n 2 d 1 = ( 1) Im (y i(t x))− − π dxn − − 1 , η > 0 9 # $ n sgn(η) = 0 , η = 0 (20) n 2 d 1  = ( 1) Re (t x + iy)− . 1 , η < 0 . − π dxn −  − 9 # $  267 Thus, Through the analysis as described in section 3.3, we have

1 + ∞ ξ y i(t x)ξ y 0 4 lim kn(t x, y)φ(t) dt iξ ξ e−| | e − dξ → , + 3 y 0 1 − | | −→ (t x) → !− !−∞ 4 5 − 1 n ∞ y 0+ n 2 d 1 2 ξ y i(t x)ξ → = lim ( 1) Re (t x + iy)− φ(t) dt ξ e−| | e − dξ 2πδ%%(t x) , y 0+ − π dxn − −→ − − → 9 1 !−∞ !− # $ 4 5 + n 1 ∞ ξ y i(t x)ξ y 0 n 2 d 1 iξe−| | e − dξ → 2πδ%(t x) , = ( 1) Re lim (t x + iy)− φ(t) dt n + −→ − − π dx y 0 1 − !−∞ 9 # → − $ 4 5 + ! ∞ y 0 n 1 ξ y i(t x)ξ → n 2 d φ(t) 1e−| | e − dξ 2πδ(t x) , = ( 1) n dt −→ − − π dx − 1 t x !−∞ 4 5 9 !− − 2 1 φ(t) where δ(x) denotes the . Thus = n!( 1)n = dt. n+1 one can reach following hypersingular integrodifferential − π 1 (t x) 9 !− − equation: where the Plemelj formula and the definition of finite part d 2 2 2 2 2 integrals have been used. Note that, when n is an odd 1 2+ 3β+ 1 3β + /8 [+%/(2+)] = − + − − integer, π (t x)3 − 2(t x)2 t x !c - − − − +% β+% p(x) 1 ∞ n n ξ ξ y+i(t x)ξ +k(x, t) φ(t)dt + φ%(x) + φ(x) = . (33) i ξ | |e−| | − dξ √2π iξ } 2 2 G !−∞ n 2 d 1 The density function φ(x) used in (33) is the strain func- = Im (t x + iy)− . − π dxn − tion, that is, φ(x) = ∂w(x, 0)/∂x with c < x < d. 9 # $ Under the case that β = 0 and + = +% = 0, integral Thus we have equation (33) has an exact solution [7]. Figure 2 shows 1 that strain is finite at the two crack-tips, which is different 1 ∞ n n ξ ξ y+i(t x)ξ dtφ(t) lim i ξ | |e−| | − dξ from the conventional linear elasticity — strain has 1/√r + 1 y 0 √2π iξ !− → !−∞ singularity. A crack surface displacement in an infinite n 1 2 d 1 nonhomogeneous plane under uniform crack surface shear = Im lim φ(t)(t x + iy)− dt n + − π dx y 0 1 − loading is shown in Figure 3, in which one can see that 9 # → !− $ dn the tangent line at the two crack-tips has infinite slope. √ = 2π n φ(x) , − dx 2.5 where the Plemelj formula is used again. 2

1.5 3.4 Strain Gradient Elasticity 1 numerical (−), analytical (o) 0.5

In addition to the choice of unknown density function, the 0 (x/a) φ underlying elasticity theory also gives rise to higher order −0.5 hypersingular kernels. For instance, in strain gradient −1 elasticity higher order of singular integral equations often −1.5 arise [1, 7, 18]. The higher order singularity is actually −2 linked to its governing PDE—a fourth order instead of −2.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 second order: x/a ∂w ∂2w ∂w +2 4w 2β+2 2 + 2w β2+2 + β = 0 , − ∇ − ∇ ∂x ∇ − ∂x2 ∂x (30) Figure 2: Numerical solution vs. closed form solution In a factored form PDE (30) can be written as for antiplane shear problem under the case β = 0 and + = +% = 0. All variables have been normalized by the ∂ ∂ half crack length a. 1 β+2 +2 2 2 + β w = 0 , (31) − ∂x − ∇ ∇ ∂x % ( % ( where + is a length parameter in the strain gradient elas- ticity theory [14]. 4 Conclusions After Fourier integral transform and asymptotic anal- ysis, the corresponding hypersingular kernel, as in (27), In this paper we have investigated how the hypersin- is gular integral equations arise either due to the choice of unknown density function or the underlying elastic- + 3β+2 + β + 2 3+2β2 iξ i+2 ξ ξ % iξ+ ξ + % + % + 1 . ity theory. The hypersingular integral is regularized by − | | − 2 2 | | 2 2+ 8 − ξ /% ( 0 | | the Hadamard finite part integral, and it leads to a very (32) stable numerical approximation.

268 1.6 [13] S. G. Mikhlin, Integral Equations and their Applications to y 1.4 Certain Problems in Mechanics, Mathematical Physics and G(x) = G eβ x 0 Technology, The MacMillan Company, New York, 1964. 1.2 [14] R. D. Mindlin, Micro-structure in linear elasticity, Arch. Ra- x 1 tion. Mech. Anal., 16, (1964) 51-78. )

0 [15] G. Monegato, Numerical evaluation of hypersingular integrals, 0.8 β = 0.1 0.25 / G J. Comput. Appl. Math., 50, (1994) 9-31. 0 0.6 0.5 [16] N. I. Muskhelishvili, Singular Integral Equations, Noordhoff 0.4 International Publishing, Groningen, The Netherlands, 1953. 1.0 w(x,0)/(ap 1.5 0.2 [17] N. I. Muskhelishvili, Some Basic Problems of the Mathematical 2.0 Theory of Elasticity, Noordhoff Ltd., Groningen, The Nether- 0 lands, 1963. 0 0.5 1 1.5 2 x / a [18] G. H. Paulino, A. C. Fannjiang, Y.-S. Chan, Gradient elasticity theory for mode III fracture in functionally graded materials – Part I: crack perpendicular to the material gradation, ASME Figure 3: Crack surface displacement in an infinite nonho- J. Appl. Mech., 70, (2003) 531-542. mogeneous plane under uniform crack surface shear load- βx ing σyz(x, 0) = p0 and shear modulus G(x) = G0e . Here a = (d c)−/2 denotes the half crack length. − 5 Acknowledgements

Chan and Feng thank the grant (Grant # W911NF-05- 1-0029) from the U.S. Department of Defense, Army Re- search Office. Organized Research Award (2006–2007) from University of Houston–Downtown is acknowledged by Chan.

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