The starry sky model of the percolation phase transition III. A new equilibrium equation and two kinds of the ‘special’ elastic nodes

The equilibrium Lame equation in two-dimensional and three-dimensional spaces can be presented as conductivity in magnetic field equation, with Lorenz force expression. This allows us to apply the perturbation theory and generalize the Lame equation, and obtain a new equilibrium equation with all responses of compression and tension (it means that compression in one direction leads to tension in the perpendicular direction, and vice versa, if the boundaries are fixed). The solution in recurrent integral form for contributions to all orders of Poisson ratio is then obtained. Zero-order solution of Lame equilibrium equation expresses the Poisson ratio as ve ∝ (p − pc)k, K = 0.3, and creates the ‘special’ elastic nodes with maximum local deformation. The first-order solution gives mean values of effective lateral displacements or forces when lateral boundaries are fixed and expresses the bulk modulus as Ke ∝ (p − pc)2t − g, where g = 0.6 is a Hall coefficient critical exponent. It creates the ‘special’ active elastic nodes, where the whole sample microrotates around. A new universality class of elasticity problem belongs to the Hall coefficient universality class, therefore the elasticity backbone coincides with the Hall coefficient backbone with the fractal dimension dfB = 2.25.