Spherical harmonics method for computing the image stress due to a spherical void

Abstract

We develop an efficient numerical method for calculating the image stress field induced by spherical voids in materials, and applied the method to dislocation-void interactions. The method is constructed based on a complete set of basis functions for the displacement potential of the elastic boundary value problem for a spherical hole, as well as the corresponding displacement, stress, and traction fields, all in terms of linear combinations of spherical harmonics. Using the fast transformation between the real and sphericalharmonics spaces provided by the SHTOOLS package, the method is more efficient than other image stress solvers such as the finite-element method. This method can be readily extended for solving elasticity problems involving inclusions and inhomogeneities, as well as contact between spheres. The tools developed here can also be useful for fast solution of differential equations with spherical boundaries beyond elasticity.

Microparticle traction force microscopy reveals subcellular force exertion patterns in immune cell–target interactions

“…. Starting from u(θ, φ) as the displacement boundary condition of the boundary value problem in small-strain linear elasticity regime, we can rapidly obtain the full solution, including the stress field σ(θ, φ) everywhere in the particle, the traction force T(θ, φ) on the surface, and the elastic energy E el using the spherical harmonics-based method 19 . This trial displacement solution guarantees the mesh nodes match the measured shape perfectly.…”

“…8. method 19 , we can solve the boundary value problem and obtain the cost function and its gradient within 0.2 s, so that the cost function can be minimized within 2 h. This would not be possible using general purpose elasticity solvers such as the finite element method (FEM).…”

“…Reference-free estimation of stresses from particle shape. Force calculations were performed with custom Python package ShElastic 19 . The particle shapes were smoothed within a 1 μm 2 window, and the edge of the traction-free boundary was dilated (Supplementary Note 4).…”

“…Using confocal microscopy, we can reconstruct the particle boundary with superresolution accuracy, allowing us to visualize cell-induced local deformations with <50 nm precision. Finally, we solve the inverse problem of inferring the displacement field and traction forces from the measured particle shape by iteratively minimizing a cost function consisting of contributions from the residual tractions on the traction-free region and the elastic energy, using a fast spherical harmonicsbased method 19 . We illustrate the potential of this MP-TFM method by revealing subcellular details of the mechanical interaction of macrophages with their targets during phagocytosis, as well as the dynamics of force exertion in the T-cell immunological synapse.…”

Superresolved microparticle traction force microscopy reveals subcellular force patterns in immune cell-target interactions

“…We started with a trial surface displacement field ሺ ߠ , ߮ ሻ , where ߠ , ߮ are colatitude and longitude respectively, from which we can rapidly obtain the full elasticity solution, including the stress field everywhere in the particle, the traction force ሺ ߠ , ߮ ሻ on the surface, and the elastic energy ‫ܧ‬ using the spherical harmonics-based method 27 .…”