Modelling of bifurcation of electro-elastic materials: Advantages and limits of the Stroh's compound matrix formulation for spherical shells

The instability and bifurcation phenomena in electro-elastic membranes and shells are receiving more attention because of their practical applications in actuators, sensors, robotics or in biological tissues used as surgical prosthetic replacements. The case where mechanical and electrical stress are applied is of particular importance since the material behaviour becomes more complex and difficult to predict. Theoretical calculations of the problem involve both mathematical methods and material's model. In this paper, the theory of small incremental electro elastic deformations superimposed on a finitely deformed shell is used to determine the possible appearance of the non-spherical shape of the shell, revealing bifurcation phenomena. The novel Stroh's compound matrix formulation is applied to solve the problem for the first time, comparing results with those previously found in the literature with the classical shooting method. Besides, the Lopez-Pamies model is taken because of the expectation of instabilities giving rise to non-spherical (aspherical) deformations and the concomitant bifurcation phenomena. Results show that, for the case of n = 0, new bifurcation bands are shown and, as a general characteristic, for numerical resolution of the equations, the compound method has a better resolution than the classical shooting method. As a limitation of the method, in the particular case of n = 1, corresponding to the first non-spherical instability, the Stroh matrix has nil determinant and, consequently, the classical shooting method of calculation is needed to solve the present problem. Thus, the Stroh's compound matrix method formulation, with the shown advantages and limitations, deserves consideration as a powerful tool for instability and bifurcation analysis.