Exact solutions for doubly curved laminated cross-ply and antisymmetric angle-ply shell substrate based bimorph piezoelectric energy harvesters

Exact solutions for the electro-elastic static response of simply supported doubly curved (DC) shell piezoelectric bimorph energy harvesters composed of laminated cross-ply or antisymmetric angle-ply composite substrate shell subjected to distributed mechanical loads have been derived. Both series and parallel connections of the piezoelectric layers of the bimorphs are considered for deriving the exact solutions. Derivation of such exact solutions is found to be possible when the piezoelectric layers are orthotropic and generally orthotropic. All linear theories of elasticity and piezoelectricity are used in orthogonal curvilinear coordinate system and a variational principle is employed to determine the boundary conditions associated with the governing equations. The electro-elastic governing equations are solved exactly to obtain the static responses of the harvesters for different shell configurations. The effects of curvature, stacking sequence of the substrate layers and connections of the piezoelectric layers on the harvesting capability of the laminated composite DC shell bimorph harvesters are investigated. It is explored from the exact solutions that the energy harvesting capability of hyperboloid DC Shell piezoelectric bimorph is maximum among the spherical, paraboloid and hyperboloid DC laminated piezoelectric shell harvesters. The expressions of the exact solutions for the DC laminated shell type piezoelectric energy harvesters derived in this paper may be treated as the benchmark solutions for verifying numerical and experimental results.