Instabilities of soft dielectrics.

The basic modern theory of nonlinear electroelasticity and its use in the formulation of constitutive laws governing the behaviour of dielectric elastomer materials was summarized in a recent review article by Dorfmann & Ogden (Dorfmann & Ogden 2017 Proc. R. Soc. A 473, 20170311 ( doi:10.1098/rspa.2017.0311 )). The theory is used, in particular, to analyse the behaviour of transducer devices such as actuators and sensors. Important considerations for the design and effective functioning of such devices are the issues of material and geometric instabilities. Following on from the above-cited work, the present paper provides a detailed account of the types of instabilities that arise for some of the geometries used in transducer devices and the theory that is adopted for the analysis of such instabilities. The theory is then used in two illustrative examples: (i) determination of instabilities of a thin electroelastic plate with flexible electrodes attached to its major surfaces, in particular comparison of the results for the so-called Hessian approach and a general incremental bifurcation analysis in respect of an equibiaxially stretched plate, with numerical results presented for a Gent electroelastic model; (ii) a general analysis of axi-symmetric bifurcation from a circular cylindrical configuration of a thin-walled tube of an electroelastic material with flexible electrodes on its curved surfaces, illustrated by numerical results for neo-Hookean and Gent electroelastic models. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.