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

QA Mathematics;QC Physics

Melnikov, Andrey ; Ogden, Raymond

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Nonlinear electroelasticity, Thick-walled electroelastic cylindrical andspherical Shells, Bifurcation and stability analysis.;   

Others  :  http://theses.gla.ac.uk/7910/1/2016melnikovphd.pdf

来源: University of Glasgow

【 摘 要 】

In this dissertation we consider some boundary value and stability problemsfor electro-active soft rubberlike materials which withstand finite deformations elastically.In the beginning we consider in detail the problem of finite deformation of a pressurizedelectroelastic circular cylindrical tube with closed ends with compliant electrodes at itscurved boundaries. Expressions for the dependence of the pressure and reduced axialload on the deformation and a potential difference between the electrodes, or uniformsurface charge distributions, are obtained in respect of a general isotropic electroelasticenergy function. To illustrate the behaviour of the tube specific forms of energy functionsaccounting for different mechanical properties coupled with a deformation independentquadratic dependence on the electric field are used for numerical purposes, for a givenpotential difference and separately for a given charge distribution. Numerical dependencesof the non-dimensional pressure and reduced axial load on the deformation are obtained forthe considered energy functions. Results are then given for the thin-walled approximationas a limiting case of a thick-walled cylindrical tube without restriction on the energyfunction. The theory provides a general basis for the detailed analysis of the electroelasticresponse of tubular dielectric elastomer actuators, which is illustrated for a fixed axial loadin the absence of internal pressure and fixed internal pressure in the absence of an appliedaxial load.Using the theory of small incremental electroelastic deformations superimposed on anelectroelastic finitely deformed body, we then look for solutions of underlying configurationswhich are different from perfect cylindrical shape of the tube. First, we considerprismatic bifurcations. We obtain the solutions which show that for neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. Thisresult is different from the pure mechanical case considered previously in Haughton and Ogden(1979), because in Haughton and Ogden (1979) prismatic bifurcation modes were foundonly 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 becomemore unstable: axisymmetric bifurcations become possible at lower values of circumferentialstretches as compared with the values of circumferential stretches found for analogousproblems solved for electromechanically indifferent materials, or equivalently, when electricfield 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 externalpressure. The solutions obtained for neo-Hookean electroelastic energy function show thatin some cases axisymmetric modes of bifurcation become possible under inflation in thepresence of electric field.