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Area of Science:

  • Solid mechanics
  • Wave propagation
  • Materials science

Background:

  • Scattering phenomena are crucial in understanding wave dynamics in structured media.
  • Lattice structures exhibit unique wave propagation characteristics.
  • Crack and rigid tips introduce stress concentrations and alter wave behavior.

Purpose of the Study:

  • To analyze the scattering of anti-plane shear waves by crack and rigid tips on a square lattice.
  • To simplify the scattering problem using structural symmetry and reduce it to scalar Wiener-Hopf equations.
  • To investigate wave incidence from both the bulk lattice and a lattice waveguide.

Main Methods:

  • Reduction of coupled equations to two scalar Wiener-Hopf equations.
  • Analysis of four geometrically reduced problems on a lattice half-plane.
  • Construction of exact solutions for different incidence scenarios.
  • Superposition of solutions to address the main scattering problem.

Main Results:

  • Exact solutions were derived for wave scattering from crack and rigid tips on a square lattice.
  • The study successfully reduced complex scattering problems to simpler Wiener-Hopf equations.
  • Wave incidence from both bulk lattice and waveguide scenarios were analyzed.

Conclusions:

  • The discrete wave paradigm from waveguide incidence is applicable to modern nanoscale electronic and thermal transport investigations.
  • The methodology provides a framework for analyzing dynamic phenomena in structured media.
  • The findings contribute to the understanding of wave localization and transport in engineered materials.