The Saint-Venant problem for general anisotropic piezoelectric cylinders with applications to smart metamaterials design

The Saint-Venant problem for general anisotropic linear piezoelectric cylinders is ad- dressed. Under the assumption of material homogeneity along the cylinder axis, the Voigt hypothesis is shown to be a necessary condition for the solution, thus allowing for the reduction of the three-dimensional piezoelectric problem to a pair of boundary-value problems on the cylinder cross section. A non-standard choice of the unknown indepen- dent fields is adopted for simplifying such a task, and the mixed-type variational structure underlying the proposed formulation is investigated. A first application deals with the analytical derivation of the piezoelectric sensor/actuator vector of a general anisotropic, homogeneous, piezoelectric cylinder, subject to mechanical/electrical loadings on its bases. Then, the solution of the relaxed Saint-Venant problem is derived in closed form in the case of a homogenous cylinder with circular cross section. On the basis of such reference solution, the accuracy and efficiency of the proposed variational formulation is proven, also in comparison to three-dimensional finite element solutions. Finally, the potentialities of the obtained results for the rational design of new smart lattice metamaterials with optimized piezoelectric properties are proven through a numerical example.