A high-order boundary integral method for surface diffusions on elastically stressed axisymmetric rods.

Area of Science:

• Materials Science
• Computational Mechanics
• Solid Mechanics

Background:

• Surface diffusion is critical in materials science, particularly for elastically stressed solids.
• Simulating singularity formation and long-time surface behavior requires accurate spatiotemporal methods.
• Axisymmetric geometries are common in applications involving surface diffusion.

Purpose of the Study:

• To develop a high-order boundary integral method for simulating surface diffusion in elastically stressed, axisymmetric solids.
• To investigate singularity formation and long-time surface evolution under elastic stress.
• To analyze the behavior of a periodic, axisymmetric elastically stressed cylinder.

Main Methods:

• A high-order boundary integral method using modified alternating quadratures and extrapolation for boundary integrals.
• A high-order integration factor method for temporal integration, reducing time-step constraints.
• A fast and accurate summation method for periodic Green's functions in isotropic elasticity.

Main Results:

• Without elastic stress, the cylinder surface pinches in finite time at the axis of symmetry, consistent with prior studies.
• With elastic stress, a geometrical singularity forms before collapse, and the corner singularity angle is estimated.
• The method achieves arbitrarily high-order accuracy in space and time.

Conclusions:

• The developed high-order boundary integral method provides accurate simulations for surface diffusion in elastically stressed solids.
• Elastic stress significantly alters the singularity formation process and timing compared to non-stressed cases.
• The findings are crucial for understanding material behavior and predicting failure in relevant applications.