On nonlinear deformations of nonlocal elastic rods

The nonlinear theory of Kirchhoff and Clebsch is extended to rods made of nonlocal materials; the motion equations are written and the balance of energy is proved. In view of application to the study of equilibrium configurations, by means of the usual assumptions of nonlocal elasticity the integro-differential equations of the theory are transformed into differential equations. Deformations of pure bending about a principal direction of inertia of rods subject to forces and couples applied at their ends are studied: equilibrium equations are integrated by means of elliptic functions, and solutions for rods made of nonlocal and classic elastic materials are compared. The curves formed in equilibrium by the axes of rods made of nonlocal materials are distinguished in inflexional and non-inflexional elastica. The equilibrium of cantilevers subject to a force applied at their free end is also examined: the results for rods of classic and nonlocal materials are discussed by regarding the deformed axial curve of a cantilever as a segment of an inflexional elastica.