Invariant solutions and constitutive laws for a nonlinear elastic rod of variable cross-section (Q1077963)

The group properties analysis is made connected with the problem of determining possible sets of exact solutions to the system of equations governing the mathematical model of homogeneous nonlinear elastic incompressible rod of variable cross-section. The cross-section of the rod is supposed everywhere symmetrical relative to the axis of the rod and to two fixed orthogonal axes that are normal to it, but the cross-sectin area is supposed to be a slowly varying function of the distance measured along the rod. By requiring the invariance under infinitesimal group of transformations for the above model, the authors are deriving the mathematical conditions characterizing classes of functional forms for the stresses and the cross-sectional area. The paper is of interest for testing numerical solutions of the governing partial equations and as asymptotic limits of other solutions of the model of homogeneous elastic rod of variable cross-section.