Yifan Wang

Yifan Wang

Postdoctoral Fellow in the Department of Materials Science and Engineering, Stanford University

Spherical Harmonic Method for Linear Elasticity with Application to Dislocation-Void Interaction

Date:

18th U.S. National Congress for Theoretical and Applied Mechanics – Jointly Organized with the Chinese Society of Theoretical and Applied Mechanics

Abstract

We develop an efficient numerical method [1] for calculating the image stress field induced by the defects in the material with spherical shape. The application of this method includes facilitating dislocation dynamics simulation of void-dislocation and precipitate-dislocation interactions. This method is also applicable to other small-deformation mechanical problems with spherical boundary conditions such as sphere contact problem and spherical shell buckling problem. The method utilizes the symmetry of sphere and expresses the image stress in spherical harmonics space. With the fast transformation and reconstruction of spherical harmonics by using SHTOOLS [2], the method is numerically efficient compare to other image stress solver such as finite-element methods. The method is based on the multipole expansion of the displacement potential functions in Papkovich-Neuber solution. We calculate and save the corresponding displacement, stress, and traction for each multipole expansion mode, and the solution to the image stress problem is developed as following steps: (1) reconstructing the traction boundary condition on the spherical boundary by the linear combination of the tractions corresponding to the multipole expansion modes; (2) the image stress is constructed by the same linear combination of the stress fields corresponding to the multipole expansion modes. With appropriate truncating of the multipole expansion modes, our method has perfect agreement with the analytical benchmark cases such as a spherical void in a tensile field [3], and the void-dislocation interaction with an infinite straight screw dislocation [4]. We also performed finite-element method (FEM) for the same problems, and for reaching sufficient accuracy, FEM is usually 20 times slower than our method.

References

  1. Wang, Yifan, Xiaohan Zhang, and Wei Cai. “Spherical harmonics method for computing the image stress due to a spherical void.” Journal of the Mechanics and Physics of Solids, 126 (2019): 151-167. (link) (code)
  2. M. A. Wieczorek, M. Meschede, I. Oshchepkov, E. Sales de Andrade, and heroxbd (2016). “SHTOOLS: Version 4.0.” Zenodo. doi:10.5281/zenodo.206114
  3. Sadd, M. H. (2009). Elasticity: theory, applications, and numerics. Academic Press.
  4. Gavazza, S. D., & Barnett, D. M. (1974). The elastic interaction between a screw dislocation and a spherical inclusion. International Journal of Engineering Science, 12(12), 1025-1043.