Analysis of the bending of a neo-Hookean electro-elastic shell of arbitrary thickness under an externally-applied hydrostatic pressure

Abstract

‎ The present study analyzes the bending of a simple electro-elastic cylindrical shell by the compound matrix method‎. The cross-section of the circular cylindrical shell is a non-circular curved shape‎, ‎with ${\mu }_1$a function of $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$B$}\right.$ and the mode number‎, ‎where $A$ and $B$ are the pre-deformation inner and outer radii of the cylindrical shell‎, ‎and ${\mu }_1$ is the ratio of the deformed inner radius to $A$ ‎. ‎In the first step‎, ‎a numerical model of the problem is developed to obtain specific differential equations‎. ‎The modeling yields a system of two Ordinary Differential Equations with three boundary conditions of the same type‎. ‎Next‎, ‎it is shown that the dependence of ${\mu }_1$ to $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$B$}\right.$ has a boundary layer structure‎. ‎Simple numerical observations were made for bifurcation conditions‎. ‎The analysis is‎, ‎in fact‎, ‎based on the variations of the inner and outer radii $A$ and $B$ ‎, ‎assuming $a={\mu }_{1}A$ and $b={\mu }_{2}B$, ‎and based on the bifurcation of ${\mu }_1$ and ${\mu }_2$ratios with respect to radius‎. ‎For this purpose‎, ‎the compound matrix method is used to show the validity of the arguments‎.

Keywords ‎
‎‎Nonlinear Electro-elasticity; ‎Compound Matrix Method; ‎Bending Bifurcation; ‎Finite Deformations; ‎Electric Field

Graphical Abstract