A boundary layer solution for the post-critical thermo-electro-mechanical stability of nonlocal-strain gradient Functionally Graded Piezoelectric cylindrical shells

Abstract

Continuum mechanics is extended to the Nonlocal (NL) and the Strain Gradient (SG) theories by incorporating the long-range molecular interactions. Nanostructures are often made of smart materials (e.g. Piezoelectric) for multifunctional attributes. The NL-SG theory based analysis/design of nanostructures offers convenience comparing the computationally expensive atomistic simulation. This study investigates the thermal post-buckling of a piezoelectric, NLSG thin cylindrical shell with thickness-wise gradation of elastic properties. The governing equations are derived following the Donnell's approach including the pre-buckling nonlinearity, large post-critical deflection and geometric imperfections. A thickness-wise cosine distribution of the electric potential is employed following the Maxwell equation. The Boundary Layer (BL) concept is employed to solve the nonlinear equations using asymptotic expansions of the regular and the BL field variables. The solution is illustrated on moderately short shells to show the prominent SG but nominal NL effects. The interactions result in conservative critical temperature but non-conservative post-critical path in contrast with the classical shell. The influence of functional gradation and the electric fields are illustrated. Increasingly sharper displacement gradient around the BL of the SG shell suggests the increasing relevance of the BL theory. The proposed solution may be useful for analysis, design and active instability control by adaptive feedback voltage.