A Note on Evaluation of Temporal Derivative of Hypersingular Integrals over Open Surface with Propagating Contour
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Abstract
The short note concerns with elasticity problems involving singular and hypersingular integrals over open surfaces, specifically cracks, with the contour propagating in time. Noting that near a smooth part of a propagating contour the state is asymptotically plane, we focus on 1D hypersingular integrals and employ complex variables. By using the theory of complex variable singular and hypersingular integrals, we show that the rule for evaluation of the temporal derivative is the same as that for proper integrals. Being applied to crack problems the rule implies that the temporal derivative may be evaluated by differentiation under the integral sign.
Keywords
Propagating crack Hypersingular integrals Differentiation with respect to parameterMathematics Subject Classification (2000)
30E20 45E05 74G701 Introduction
Using singular and hypersingular integrals and boundary integral equations (BIE) has proved to be a highly efficient means for solving problems of fluid and solid mechanics (see, e.g., [2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 15, 18]). The modern theories of hypersingular integrals and HBIE, both real and CV, are comprehensive when the boundary of the region of integration is fixed. However, there have arisen new computational problems involving propagating surfaces (see, e.g., [21]), which require formulae for temporal derivative of singular and hypersingular integrals. Such formulae are also needed for the sensitivity analysis applied to error estimation of the boundary element method [8]. In the case of nonsingular integrals, the differentiation rules were given in [19]. They have been employed in [8] for obtaining the derivative of a singular integral over a surface of a 3D domain when points of the surface move in such a way that the initial domain stays globally unchanged (the changes in positions of the surface points occur in the tangential direction). This case is of prime significance when studying how the change of the position of a collocation point influences the value of a singular integral.
In the present paper, we are interested in another case, when the surface is open and the positions of its points behind a propagating contour do not change, while the contour and density change in time. We have come across such a problem when studying hydraulic fracturing widely used for stimulation of oil, gas and heat production (see, e.g., [1]). Then employing the temporal derivative of the hypersingular integral might notably facilitate numerical modeling of the fracture propagation in time. The main difficulty when obtaining the differentiation rule for this case is caused by the moving boundary rather than by the change of the density in time. Indeed, for a fixed contour, the common definition of the principal value or finitepart integral as the limit after exclusion a small εvicinity of a singular point, leads (under physically sound conditions on the smoothness of the surface, contour and density) to the conclusion that the differentiation may be performed under the integral sign. Consequently, for a moving crack front, we may fix a contour close to the front at the time instant considered and represent the integral as the sum of that with the fixed boundary and the integral over the thin strip between the fixed contour and the front. Thus the difficulty actually refers to differentiation of the integral over the thin strip. The latter integral involves the asymptotic behaviour of the density near the boundary. In applied problems, the asymptotic behaviour is asymptotically plane. This implies that to obtain the differentiation rule, it is reasonable to focus on 1D singular and hypersingular integrals of the plane potential and elasticity problems. Then the CauchyRiemann conditions for a harmonic function suggest using holomorphic functions having derivatives of an arbitrary order. This property is of key significance to connect the limiting values of the Cauchy type integral and Hadamard type integral, when the field point goes to the contour (surface) of integration, with the values of density and direct (principal, finitepart) values of Cauchy and Hadamard integrals. In real variables this beneficial property is reached by using the distribution theory (see, e.g., [16]).
Below we employ the advantages of the CV holomorphic functions in the CV variable and the theory of CV singular [17, 18] and hypersingular [13, 14] integrals to derive the needed rule for differentiation with respect to a parameter. Higher order hypersingular integrals are included into the rule because of their presence in efficient quadrature rules used in numerical solutions of hypersingular integrals (see, e.g., [13]). The evident extensions to singular and hypersingular integrals over an open surface with propagating contour are sketched in comments at the ends of Sects. 3 and 4.
2 Starting Definitions

the functions x(γ) and y(γ) are continuous on the closed interval [γ _{ a },γ _{ b }],

they have continuous derivatives x′(γ) and y′(γ) on the open interval (γ _{ a },γ _{ b }),

the derivatives are not zero simultaneously, that is x′(γ)^{2}+y′(γ)^{2}>0 for γ∈(γ _{ a },γ _{ b }),

there are no branchpoints on the arc what means that the simultaneous equalities x(γ _{1})=x(γ _{2}) and y(γ _{1})=y(γ _{2}) imply that γ _{1}=γ _{2}.
We accept these conditions and call such an arc a smooth arc.
In further discussion, the positions of start and end points may change depending on a real parameter α. (In applied problems the parameter is commonly the time.) Then γ _{ a }=γ _{ a }(α), γ _{ b }=γ _{ b }(α), a=a(α), b=b(α). The curve is smooth for each value of α.
Let a CV function g(τ) be prescribed at points of the arc (a,b). We assume it Holder continuous at (a,b), that is [18] there exist nonnegative numbers A and μ≤1 such that g(τ _{1})−g(τ _{2}) ≤Aτ _{1}−τ _{2}^{ μ } for any τ _{1},τ _{2}∈(a,b).
3 Formula for Differentiation of a Hypersingular Integral with Respect to a Parameter
Herein, we have changed the order of differentiation on the l.h.s. what is justified under the conditions accepted.
We have proved the theorem expressing the rule of differentiation of a hypersingular integral with respect to a parameter.
Theorem
For a smooth arc(a,b) with a(α) and b(α) being Holder continuous in a parameter α and for a density g(α,τ) having (k−1)th Holder continuous derivative with respect to τ and Holder continuous derivative with respect to α, the derivative of a hypersingular integral I _{ k }(α,t) with respect to the parameter α has the form (10) reproducing the common rule for proper integrals.
Remark 1
Similar rule holds for 2D singular and hypersingular integrals over an open surface with a propagating front.
4 Extension to Densities with Derivatives Having PowerType Singularity at Arc Tips
In applied problems concerning with cracks, k=2 and α has the meaning of the time. Commonly, the integral on the l.h.s. of (7) is proportional to the netpressure on crack surfaces, the density g(α,τ) is the fracture opening and the derivatives db/dα and da/dα express the speeds, with which the fracture front propagates. According to (10), the influence of the speeds on the rate of the pressure change strongly depends on the values g(α,a) and g(α,b) of the opening at the points of the front a and b. Usually, near a point c of the front, the opening tends to zero as (c−τ)^{ γ }, where Re(γ)>0. In particular, in fracture mechanics, commonly γ=0.5 (see, e.g., [20]); in problems of hydraulic fracture, propagating in the viscosity dominated regime, γ=2/3 (see, e.g., [22]); for the leakoff dominated regime, γ=5/8 (see, e.g., [12]). Hence, we need to extend the theorem to the case when near an edge point c (c=a or c=b) the density is of the form g(α,τ)=(c−τ)^{ γ } g _{ γ }(α,τ), where Re(γ)>0 and the function g _{ γ }(α,τ) meets the conditions of the theorem. Note that g(α,c)=0.
Extended Theorem
For a density having representation g(α,τ)=(c−τ)^{ γ } g _{ γ }(α,τ) near start (c=a) and end (c=b) points, the theorem holds for points t within an open arc (ab).
It appears (see Appendix) that when the distance d between the point t and a tip c goes to zero, the integral (12) behaves as O(d ^{−k+γ }), when γ≠1/2; it is nonsingular, when γ=1/2.
Remark 2
In applied problems for 2D surfaces, the asymptotic behaviour of the crack opening near a smooth part of a contour is the same as in a planestrain problem. Consequently, integration under the integral sign is possible in these problems, as well.
5 Summary

the temporal derivative of singular and hypersingular integrals may be evaluated by the common rule for proper integrals;

it is possible to evaluate the temporal derivative under the integral sign when either the boundary is fixed, or the density is zero on the moving boundary; in these cases, the singular and hypersingular boundary integral equations keep their from for temporal derivatives of physical quantities;

near a smooth part of a propagating boundary, the temporal derivative of a hypersingular integral of order k asymptotically behaves as O(d ^{−k+γ }), when the density asymptotically behaves as O(d ^{ γ }) with 0<γ<1, γ≠1/2 and d being the distance from the boundary. The temporal derivative is nonsingular near a smooth part of the boundary if γ=1/2.
Notes
Acknowledgements
Authors gratefully acknowledge the support of the European Research Agency (FP7PEOPLE2009IAPP Marie Curie IAPP transfer of knowledge programme, Project Reference # 251475).
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