by @MiroK. It works like a magic in my project, but as someone without solid mechanics background, I’m wondering if anybody understands how this is come up and share some insights here, for me and anyone who need it in the future.

Really appreciate any explanation/reference etc
Victor

Hi, the 6 vector fields that are in the nullspace of the linear elasticity operator are the deformations u for which (\nabla u+(\nabla u)^T)=0 and \nabla\cdot u = 0. The latter term relates to the change of volume due to u, i.e. you are looking for some deformation which preserves volume. Intuitively, translations and rotations are (only) such vector fields; the body which is moved and/or rotated keeps its volume, right? In 3d you have 3 translations (around each of the axes) and 3 rotations (around each of the axes). Z_{rot} are rotations around x, y, z-axis .