On the bending problem of a polar linearly elastic cantilever

This communication aims to initiate an investigation regarding the linearly elastic bending response of a cantilevered bar that exhibits features of polar material behaviour, while it may also be influenced by anisotropy features that are due to fibre presence. The existing exact elasticity solution of the bending problem of a corresponding isotropic cantilever is currently confined within the bounds of the non-polar linear elasticity, and refers only to the stress analysis part of the problem. Complete solution of that problem still requires determination of the corresponding displacement field which, while depends on the shape of the bar cross-section, is also needed for the solution of the considered polar material version of the problem. Accordingly, the present part of this investigation initially extends the existing non-polar elasticity solution by including material symmetries that represent the class of material orthotropy, and completes it by determining the corresponding displacement field for an orthotropic cantilever with circular cross-section. In the special case of material isotropy, it then achieves to complete the corresponding non-polar elasticity solution, by (i) providing the relevant displacement field, and (ii) studying the bending response of the corresponding isotropic polar material cantilever. Nevertheless, substantial similar progress is also made in a case of polar transverse isotropy, where polar material response of a bent cantilever bar is anticipated due to a family of embedded fibres that possess bending stiffness.