Reissner Mixed Theorem Applied to Static Analysis of Piezoelectric Shells†

Abstract

This article addresses the static analysis of multilayered shells with embedded piezoelectric materials. The Reissner Mixed Variational Theorem (R MVT) is applied to derive governing differential equations of doubly curved shells by referring to Gibbs free energy G and Electric Gibbs energy G2. Interlaminar continuity of transverse stress (both shear and normal components) is a priori fulfilled by RMVT applications which permits to assume two independent fields for displacement and transverse stress variables. Two-dimensional approximations in the shell thickness direction z are introduced by application of Unified Formulation and a number of advanced mixed theories are extended to piezoelectric shells. Both Layer-Wise and Equivalent Single Layer models have been addressed. Up-to-forth order expansion in z have been implemented in the numerical investigation. Closed form solutions have been obtained in the case of simply supported, orthotropic shells subject to harmonic distribution of electric and mechanical loadings. Classical shell theories with only displacement unknowns have been included in the numerical discussion for comparison purpose. Numerical analysis have first considered plate and shell problems for 3D exact solution has been provided. Two benchmarks have been then proposed which refer to a shell panel and to a cylindrical shell, respectively. Both actuator and sensor case have been analyzed. The obtained results have confirmed the superiority of RMVT application with respect to classical shell theories as well as the importance to refer to Layer-Wise modeling if accurate description of mechanical and electrical field is required in the various layers.