Bifurcation of thick-walled electroelastic cylindrical and spherical shells at finite deformation

Melnikov, Andrey (2017) Bifurcation of thick-walled electroelastic cylindrical and spherical shells at finite deformation. PhD thesis, University of Glasgow.

Abstract

In this dissertation we consider some boundary value and stability problems
for electro-active soft rubberlike materials which withstand finite deformations elastically.
In the beginning we consider in detail the problem of finite deformation of a pressurized
electroelastic circular cylindrical tube with closed ends with compliant electrodes at its
curved boundaries. Expressions for the dependence of the pressure and reduced axial
load on the deformation and a potential difference between the electrodes, or uniform
surface charge distributions, are obtained in respect of a general isotropic electroelastic
energy function. To illustrate the behaviour of the tube specific forms of energy functions
accounting for different mechanical properties coupled with a deformation independent
quadratic dependence on the electric field are used for numerical purposes, for a given
potential difference and separately for a given charge distribution. Numerical dependences
of the non-dimensional pressure and reduced axial load on the deformation are obtained for
the considered energy functions. Results are then given for the thin-walled approximation
as a limiting case of a thick-walled cylindrical tube without restriction on the energy
function. The theory provides a general basis for the detailed analysis of the electroelastic
response of tubular dielectric elastomer actuators, which is illustrated for a fixed axial load
in the absence of internal pressure and fixed internal pressure in the absence of an applied
axial load.
Using the theory of small incremental electroelastic deformations superimposed on an
electroelastic finitely deformed body, we then look for solutions of underlying configurations
which are different from perfect cylindrical shape of the tube. First, we consider
prismatic bifurcations. We obtain the solutions which show that for neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This
result is different from the pure mechanical case considered previously in Haughton and Ogden
(1979), because in Haughton and Ogden (1979) prismatic bifurcation modes were found
only for an externally pressurised tube. Second, we consider axisymmetric bifurcations,
and we obtain results for neo-Hookean and Mooney-Rivlin electroelastic energy functions.
Our solutions show that in the presence of an electric field the electroelastic tube become
more unstable: axisymmetric bifurcations become possible at lower values of circumferential
stretches as compared with the values of circumferential stretches found for analogous
problems solved for electromechanically indifferent materials, or equivalently, when electric
field is not present.
Within similar lines we consider the bifurcation of a thick-walled electroelastic spherical shell with compliant electrodes at its curved boundaries under internal and external
pressure. The solutions obtained for neo-Hookean electroelastic energy function show that
in some cases axisymmetric modes of bifurcation become possible under inflation in the
presence of electric field.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Nonlinear electroelasticity, Thick-walled electroelastic cylindrical and spherical Shells, Bifurcation and stability analysis.
• Subjects: Q Science > QA Mathematics; Q Science > QC Physics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Ogden, Professor Raymond
• Date of Award: 2017
• Depositing User: Mr Andrey Melnikov
• Unique ID: glathesis:2017-7910
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 03 Feb 2017 09:41
• Last Modified: 27 Feb 2017 08:41
• URI: https://theses.gla.ac.uk/id/eprint/7910