Static plane-strain deformation of transversely isotropic magneto-electro-elastic and layered cylinders to general surface loads

Abstract

• The separation of variables and eigenfunction expansion methods are first introduced. • Static response of layered anisotropic cylinders made of multi-phase materials is solved analytically. • The present solutions can be reduced to the well-known results in layered isotropic cylinders. • Numerical examples are presented to demonstrate the dependence of the field responses on material anisotropy and layering. • Coupling and interaction among elastic, piezoelectric and piezomagnetic fields are further studied. Anisotropic and layered cylinders are important composite structures; however, their system of governing equations is usually solved numerically due to the complicated geometry and material anisotropy involved. In this paper, we analytically solve the plane-strain equations for general static deformation of a cylindrically anisotropic, layered magneto-electro-elastic (MEE) cylinder. We assume that the layers are perfectly bonded at the interfaces. We solve the equations through separation of variables and eigenfunction expansion. Results for each mode shape (2π periodic) are solved independently. Because the eigenspace of the mode shapes is the set of all continuous functions on the interval, any continuous loading can be applied and the corresponding solution can be found analytically through superposition of the mode-shape results. To check our formulation, we consider a cylinder with two isotropic-elastic layers under simple radial loading and reproduce the known, exact results. Then, we compare our formulation to an FEA solution for a layered piezo-electric (PE) cylinder. Finally, we apply a radial stress to three comparable MEE cylinders (one uniform MEE cylinder and two layered cylinders made of alternating piezo-electric (PE) and piezo-magnetic (PM) materials). Deformation and stress amplitudes are plotted for the first six mode shapes of each cylinder as benchmarks for further reference.