# A Note on Evaluation of Temporal Derivative of Hypersingular Integrals over Open Surface with Propagating Contour

## TL;DR

This paper discusses evaluating time derivatives of hypersingular integrals over moving open surfaces, like cracks, using complex variables.

## Contribution

The paper shows that the temporal derivative rule for hypersingular integrals is the same as for proper integrals.

## Key findings

- The temporal derivative can be evaluated by differentiating under the integral sign for crack problems.
- The state near a smooth part of a propagating contour is asymptotically plane.
- Complex variable theory simplifies the evaluation of 1D hypersingular integrals.

## 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.