A variational formulation for a boundary value problem considering an electro-sensitive elastomer interacting with two bodies R. Bustamante * Universidad de Chile, Departamento de Ingenierı ´a Mecánica, Beaucheff 850, Santiago Centro, Santiago, Chile article info Article history: Received 14 April 2009 Received in revised form 19 May 2009 Available online 30 May 2009 Keywords: Electro-elasticity Variational formulations Electro-sensitive elastomers Boundary conditions abstract We present a variational formulation for an electro-elastic body in contact with two semi- infinite rigid bodies, which are electric conductors and have a distribution of free charge. These three bodies are surrounded by free space, where far away we have a given electric displacement and an electric potential on disjoint surfaces. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction In recent years there has been an increasing interest in the developing of theories for electro-elastic interactions consid- ering finite deformations (Dorfmann and Ogden, 2005, 2006; Fosdick and Tang, 2007; Goulbourne et al., 2005; McMeeking and Landis, 2005; Suo et al., 2008; Vu et al., 2007). One of the applications of these theories is the modelling of electro-sen- sitive (ES) elastomers (Bossis et al., 2001). In this paper, we explore the problem of proposing a simple variational formulation for the case we have a body made of an electro-sensitive elastomer, interacting with two rigid bodies (electric conductors), all of them surrounded by free space. The reason we are interested in studying a problem like this, is connected with the fact that Maxwell equations must hold everywhere, not only for an electro-elastic body under consideration, but also for the whole surrounding space (Kovetz, 2000). This has prompted some researchers to consider in their formulations bodies surrounded by free space (Dorfmann and Ogden, 2005, 2006; Bustamante et al., 2007). However, as pointed out by the author in Bustamante (2009), to consider a body totally surrounded by free space may not be a good approximation of the ‘real’ situation. The application of surface traction means our ES elastomeric body should be in contact with other bodies, and, therefore, we should consider also those bodies in our analysis. The bodies that interact with our ES elastomeric body may deform and may also be interacting with other bodies. For simplicity we just consider one case, in which an ES elastomeric body is perfectly attached to two rigid semi-infinite bodies, which are perfect electrical conductors with a distribution of free charge on their surfaces. These bodies may rotate and move rigidly. In Section 2, we review briefly the main equations of the theory of electro-elastic interactions with finite elastic deformations. In Section 3, we give more details about the class of boundary value problem we aim to study. In Section 4, we show a variational formulation for the above model. Finally, in Section 5 we present some final remarks. 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.05.009 * Tel.: +56 2 9784597; fax: +56 2 6896057. E-mail address: rogbusta@ing.uchile.cl Mechanics Research Communications 36 (2009) 791–795 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom