Lie group analysis and plane strain bending of cylindrical sectors for compressible nonlinearly elastic materials

The plane strain bending of cylindrical sectors for two important classes of homogeneous, isotropic, compressible, nonlinearly elastic materials is considered. In contrast to the traditional approach to this problem, a double semi-inverse approach is adopted here. The strain-energy densities and the deformation fields considered each contain an arbitrary function. The deformation field is completely determined from the equations of equilibrium (as is usual) but additionally here the strain-energy function is completely determined on using Lie's linearization theorem. This theorem gives necessary and sufficient conditions for a second-order ODE to be linearizable, i.e. for a second-order equation to have a solution of the form y = ax + b, where a, b are constants and x, y are transformed variables. For harmonic materials, it has been shown in the literature that a quadrature solution can be developed. Due to the restrictive nature of an invertibility assumption inherent in this approach, however, it is shown here that very few closedform solutions can be found in this way. The use of Lie's linearization theorem does lead to some new solutions for particular harmonic materials. This method does not, however, provide any new solutions for Varga materials. When the linearization theorem fails to determine the arbitrary function in the strainenergy density, invariance techniques are used. A new parametric solution for a specific Varga material is determined in this way.