Constitutive relation for exact model of a Cosserat elastic rod

There are wide backgrounds of applications of a thin elastic rod. For an exact model of a Cosserat elastic rod which are not only bending and torsion but also shearable and extensible, constitutive relation for principal vector and principal moment on a cross section versus displacements of the section with arc coordinate are derived from Hooke´s low. It shows that derivative of generalized coordinates in bending components of curvature-twisting vector and axial components of strain vector of center of a cross section which are all composed of linear strain is about arc coordinate before deflection of the rod. And the one in torsion components of curvature-twisting vector and shear components of strain vector of center of a cross section which are all composed of shearing strain is about arc coordinate after deflection of the rod. In the senses of first-order approximation, the principal vector is proportional to the strain vector of a point in the center line, and the principal moment is also proportional to the curvature-twisting vector about intrinsic arc coordinate.