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<p>As evidently demonstrated by Professor Mark Kachanov, micromechanics of averages cannot predict fracture phenomena, which are governed by <i>extreme</i> rather than <i>average</i> values of physical fields. The study of extremes involves <i>tails</i> of statistical distributions. To date, research in this area has been limited to 2D problems. Extensions to 3D problems have been hindered primarily by the presence of finite part (Hadamard) integrals in the boundary integral equations, whose solutions define the key characteristics of the field strength—the field/stress intensity factors (FIF). This difficulty has now been overcome by the tangent plane approach, which simplifies the evaluation of finite-part integrals to the level of proper integrals.</p><p>Meanwhile, the few publications, addressing FIFs at surface edges in 3D, do not provide reliable methods for accurate evaluation. Developing such methods remains an important task, requiring comparison of results from repeated numerical experiments with accurate or exact solutions to bench-mark problems. These benchmarks should be representative and correspond to 3D cases while maintaining minimal computational complexity to reduce time cost.</p><p>The present work addresses this need by providing accurate numerical and exact analytical values of FIFs for an impermeable arc in an infinite plane under a prescribed flux at infinity. These results serve as benchmarks for comparison with the FIFs at the midpoint of an impermeable cylindrical surface whose cross-section corresponds to the arc.</p>

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Bench-mark potential problems for plane with impermeable arc

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Aleksandr Linkov,
  • Ewa Rejwer-Kosińska,
  • Liliana Rybarska-Rusinek,
  • Anastasia Dobroskok

摘要

As evidently demonstrated by Professor Mark Kachanov, micromechanics of averages cannot predict fracture phenomena, which are governed by extreme rather than average values of physical fields. The study of extremes involves tails of statistical distributions. To date, research in this area has been limited to 2D problems. Extensions to 3D problems have been hindered primarily by the presence of finite part (Hadamard) integrals in the boundary integral equations, whose solutions define the key characteristics of the field strength—the field/stress intensity factors (FIF). This difficulty has now been overcome by the tangent plane approach, which simplifies the evaluation of finite-part integrals to the level of proper integrals.

Meanwhile, the few publications, addressing FIFs at surface edges in 3D, do not provide reliable methods for accurate evaluation. Developing such methods remains an important task, requiring comparison of results from repeated numerical experiments with accurate or exact solutions to bench-mark problems. These benchmarks should be representative and correspond to 3D cases while maintaining minimal computational complexity to reduce time cost.

The present work addresses this need by providing accurate numerical and exact analytical values of FIFs for an impermeable arc in an infinite plane under a prescribed flux at infinity. These results serve as benchmarks for comparison with the FIFs at the midpoint of an impermeable cylindrical surface whose cross-section corresponds to the arc.