The Green's functions for peridynamic non-local diffusion

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.