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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
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Microcracking in concrete refers to the tiny cracks that can form within the material even before any external load is applied. These microcracks typically occur at the interface between the coarse aggregate and the hydrated cement paste, often as a result of differential volume changes prompted by variations in stress-strain behavior, as well as thermal and moisture movement. Initially, these microcracks remain stable and do not grow substantially until the concrete is stressed to about 30...
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Non-structural cracks are primarily of three types: plastic, early-age thermal, and drying shrinkage cracks. Plastic cracks are further classified into plastic shrinkage cracks and plastic settlement cracks.
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Scattering on a square lattice from a crack with a damage zone.

Basant Lal Sharma1, Gennady Mishuris2

  • 1Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India.

Proceedings. Mathematical, Physical, and Engineering Sciences
|April 10, 2020
PubMed
Summary

This study presents an exact solution for wave scattering by a cracked lattice with a damaged zone. The novel method simplifies complex problems, enabling precise analysis of wave propagation and scattering effects.

Keywords:
Wiener–Hopf methodcrackdamage zonediffraction

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Area of Science:

  • Solid Mechanics
  • Wave Propagation
  • Materials Science

Background:

  • Wave scattering by cracks is a fundamental problem in solid mechanics.
  • Previous solutions exist for atomically sharp cracks but not for damaged zones.
  • Modeling damaged zones introduces significant mathematical complexity.

Purpose of the Study:

  • To develop an exact analytical solution for wave scattering by a semi-infinite crack in a lattice with a damaged zone.
  • To simplify the complex mathematical formulation of wave scattering in damaged materials.
  • To provide a method applicable to arbitrarily distributed stiffness in damaged links.

Main Methods:

  • Modeling a partially damaged zone ahead of the crack tip with variable stiffness.
  • Reducing a complex matrix kernel problem to a scalar one using an original technique.
  • Solving an auxiliary linear system of N x N equations to obtain the exact solution.

Main Results:

  • An exact solution is constructed for wave scattering by a cracked lattice with a damaged zone.
  • The method effectively reduces complex problems to a solvable scalar form.
  • Numerical examples and asymptotic approximations of the scattered field are presented.

Conclusions:

  • The proposed technique offers an effective way to solve intricate wave scattering problems in damaged lattices.
  • This method provides a pathway to exact solutions where previous approaches were limited.
  • The findings are valuable for understanding wave behavior in materials with crack-tip damage.