A coupled hp‐finite element scheme for the solution of two‐dimensional electrostrictive materials

Abstract

As part of the ongoing research within the field of computational analysis for the coupled electro-magnetomechanical response of smart materials, the problem of linearised electrostriction is revisited and analysed for the first time using the framework of hp-finite elements. The governing equations modelling the physics of the dielectric are suitably modified by introducing a new total Cauchy stress tensor (A. Dorfmann and R.W. Ogden. Nonlinear electroelasticity. Acta Mechanica, 174:167-183, 2005), which includes the electrostrictive effect and a staggered partitioned scheme for the numerical solution of the coupling phenomena. With the purpose of benchmarking numerical results, the problem of an infinite electrostrictive plate with a circular/elliptical dielectric insert is revisited. The presented analytical solution is based on the theoretical framework for two-dimensional electrostriction proposed by Knops (R.J. Knops. Twodimensional electrostriction. Quarterly Journal of Mechanics and Applied Mathematics, 16:377-388, 1963) and uses classical techniques of complex variable analysis. Our presentation, to the best of our knowledge, provides the first correct closed form expression for the solution to the infinite electrostrictive plate with a circular/elliptical dielectric insert, correcting the errors made in previous presentations of this problem. We use this analytical solution to assess the accuracy, efficiency and robustness of the hp-formulation in the case of nearly incompressible electrostrictive materials. 1159where Maxwell's equations, which govern electromagnetic phenomena, reduce to the electrostatic model in which the electric field can be described by the gradient of a scalar potential, and the Navier-Stokes equations, which describe the mechanical-fluid behaviour, reduce to the equations of linearised elasticity in the case of small strains. The coupling being through nonlinear constitutive laws describing the electrostrictive materials.Under the assumption of very small strains, so that the effect of the electric field on the deformation of the dielectric can be ignored, Knops [3] has presented a theoretical framework for analysing two-dimensional problems in electrostriction. With the use of classical techniques of complex variable analysis, this framework has been subsequently used to obtain analytical solutions to a number of benchmark problems including an infinite electrostrictive plate with a rigid circular/elliptical dielectric insert [3][4][5][6]. Clearly, this type of analytical solution can only be obtained for simple geometries and therefore has only limited practical application. Nevertheless, such solutions are extremely useful for benchmarking computation procedures for electrostriction. We therefore revisit the classical solution of an infinite electrostrictive plate with a rigid circular/elliptical dielectric insert and correct the mistakes found in previous presentations and present, to the best of our knowledge, the first correct solution to this problem.Our computational ap...

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