Variational analysis of deformation of a compressible rod with initial bending under induced anisotropy

The longitudinal bending of a flexible rectilinear rod having an initial technological imperfection in the form of an initial die is considered. The inhomogeneity of the stress state in the bent state leads to constrained deformations causing variability of the elastic characteristics of the material. To obtain the equation of the rod in the deflected state, the theory of deformation of bodies in inhomogeneous stress fields with induced anisotropy of properties proposed earlier by the authors is used. The solution is based on the variational analysis of the deflections of the rod using the Bubnov — Galerkin method. The growth of the maximum deflection at the steps of successive loading with increasing force is analysed. Three variants of initial curvatures, including vanishing small curvature, are considered. Approximate integration is performed using simple basis functions. Regardless of the degree of initial curvature and the choice of the approximating function, a noticeable increase in the compressive force corresponding to a significant increase in deflections compared to the bifurcation approach is found.