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The Green's functions for peridynamic non-local diffusion.

L J Wang1, J F Xu2, J X Wang3

  • 1State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science , College of Engineering, Peking University , Beijing 100871, People's Republic of China.

Proceedings. Mathematical, Physical, and Engineering Sciences
|October 8, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a Green's function method for peridynamic non-local diffusion models, offering a new approach to solving diffusion problems with enhanced accuracy and applicability to various scenarios.

Keywords:
Green’s functionheat conductionnon-local diffusionperidynamics

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Area of Science:

  • Physics
  • Applied Mathematics
  • Materials Science

Background:

  • Classical diffusion models rely on local gradients, limiting their accuracy for phenomena with long-range interactions.
  • Peridynamic diffusion models offer a non-local approach, but efficient solution methods are needed.

Purpose of the Study:

  • To develop and validate a Green's function method for solving peridynamic non-local diffusion models.
  • To establish the convergence of peridynamic solutions to classical solutions as non-local effects diminish.
  • To apply the method to a practical problem and compare results with classical theories.

Main Methods:

  • Developed a Green's function method tailored for peridynamic non-local diffusion.
  • Utilized Fourier transforms to derive Green's functions for unsteady and steady diffusion in infinite domains.
  • Applied analytical solutions to an infinite plate heated by a Gaussian source.

Main Results:

  • General solutions of the peridynamic model are expressed as functionals of Green's functions.
  • Demonstrated convergence of peridynamic solutions to classical differential solutions as non-local length approaches zero.
  • Peridynamic model predicts a lower rate of variation in field quantities compared to classical theory, aligning with experimental data.

Conclusions:

  • The Green's function method provides an effective analytical solution for peridynamic non-local diffusion.
  • The peridynamic model offers a more realistic representation of diffusion phenomena, especially at larger scales.
  • The developed method is broadly applicable to various diffusion-type problems.