title: A Note on Evaluation of Temporal Derivative of Hypersingular Integrals over Open Surface with Propagating Contour.
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.
authors: Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland.
authors: Dawid
authors: D
authors: Jaworski
authors: Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland.
authors: Aleksandr
authors: A
authors: Linkov
authors: Rzeszow University of Technology, Powstancow Warszawy 12, 35-959 Rzeszow, Poland.
authors: Liliana
authors: L
authors: Rybarska-Rusinek
keywords: Differentiation with respect to parameter; Hypersingular integrals; Propagating crack