A new procedure based on constructing orthonormal tensor basis using the form-invariant expressions which can easily be extended to any tensor of rank n. A new decomposition, which is not in literature, of the stress tensor is presented. An innovational general form and more explicit physical property of the symmetric fourth rank elastic tensors is presented. A new method to measure the stiffness and piezoelectricity in the elastic fiber reinforced composite and piezoelectric ceramics materials using the norm concept on the crystal scale. This method will allow to investigate the effect of fiber orientaion, number of plies, material properties of matrix and fibers, and degree of anisotropy on the stiffness of the structure. The results are compared with those available in the literature for semiconductor compounds, piezoelectric ceramics and reinocored composite materials.