Recently some solids were classified as kinking nonlinear elastic (KNE) solids because of plastic anisotropy. The KNE solids exhibit large mechanical hysteresis due to the reproducible growth and shrinkage of incipient kink bands (IKBs). IKBs are the precursor of regular kink bands. The stress-strain curves of KNE solids can be described by 4 parameters. In this book a microscale model based on fully reversible dislocation motion is proposed to relate the four parameters. A mathematical method based on Preisach-Mayergoyz model (P-M model) was also developed to calculate the strain of KNE solids from their stress history. Select MAX phases (Ti3SiC2, Ti2AlC and some of their solids solutions) and the hexagonal metals (Ti, Mg, Co) were tested. Their kinking nonlinear elastic deformations were studied as a function of grains size, porosity, deformation history and texture. IKB theory successfully explains the damping, microyielding and initial deformation of hexagonal metals, which are not fully understood or misunderstood for a long time. Based on this work there is little doubt that incipient kink bands constitute the last piece in the deformation-of-solids puzzle.