A 2-D coupled BEM–FEM simulation of electro-elastostatics at large strain D.K. Vu, P. Steinmann * Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstrasse 5, 91058 Erlangen, Germany article info Article history: Received 12 October 2009 Received in revised form 30 November 2009 Accepted 3 December 2009 Available online 22 December 2009 Keywords: Coupled BEM–FEM analysis Nonlinear coupling Nonlinear electrostatics Nonlinear elastostatics abstract The numerical simulation of nonlinear electro-elastostatics is considered in this work using the coupled boundary and finite element method. The objective of the work is to properly simulate the deformation of an electroelastic body undergoing large deformation subjected to electric stimulations in the case, where the surrounding space has significant influence on the electric field inside the body. Finite elements are used to model the nonlinear electroelastic body in which both mechanical nonlinearity and electrical nonlinearity are taken into account. Boundary elements are used to model the surrounding space and account for the large deformation of the boundary of the body. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The simulation of nonlinear electro- and magneto-elastics [1–9] has been an interesting subject for the last few years due to its application in the analysis of smart materials, which exhibit large displacement and change their mechanical behavior in response to electric or magnetic stimulations [10,11]. An example of these materials is the so-called electronic electroactive polymers. Prom- ising applications of electronic electroactive polymers can be found in the development of artificial muscles, where electronic electro- active polymers are used as actuators [10]. When the applied elec- tric field can be considered as static and the response of the material is elastic, the behavior of electronic electroactive poly- mers can be simulated using the theory of nonlinear electro-elasto- statics. In an attempt to model the behavior of these materials, the problem of nonlinear electro-elastostatics was addressed in a re- cent work [12], in which, based on the basic equations of nonlinear electroelasticity with large deformation and nonlinear electric fields, a variational formulation was built, linearized and discret- ized using the finite element method. One assumption used in [12] is that the contribution of the vacuum to the energy function is small compared with the energy stored inside the elastic body under consideration. This leads to the neglect of the surrounding space and the use of the finite element method. In the case where the contribution of the surrounding space is significant and must be accounted for, the use of the boundary element method in con- nection with the finite element method is suitable. In this work, the finite element method is used to model the nonlinear elastic body and the boundary element method is used to model the sur- rounding space. In the next sections, the basic equations in nonlin- ear electro-elastostatics are presented based on the works [4,12– 14]. Besides, a boundary element formulation for the simulation of electrostatics is also recalled. Next, the discretization and linear- ization of the problem are introduced. For illustration, numerical examples are presented. 2. Governing equations in nonlinear elastostatics We consider here a body made of elastic material undergoing large deformation. The undeformed or reference configuration of the body is denoted by B 0 . In the undeformed configuration B 0 , the position vector of a point P (particle) of the body is denoted by X. The deformed configuration of the body is denoted by B t . In the deformed configuration B t , the position vector of the point P is denoted by x and the deformation map that maps X to x is de- noted x ¼ uðXÞ. The deformation at every point inside the body is characterized by the deformation gradient tensor F defined as: F ¼ r X u (see Fig. 1). In reference to the undeformed configuration B 0 , the Piola stress tensor P is defined as: P :¼ @ F W 0F ; ð1Þ where W 0F ¼ W 0F ðF Þ is the internal potential energy density per unit volume of the undeformed body. In reference to B 0 , the bal- ance equation of linear momentum and the boundary condition for P are written in the form: r X  P þ b 0 ¼ 0 in B 0 and P  N ¼ t 0 on @B 0 ð2:1-2Þ 0045-7825/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2009.12.001 * Corresponding author. E-mail address: paul.steinmann@ltm.uni-erlangen.de (P. Steinmann). Comput. Methods Appl. Mech. Engrg. 199 (2010) 1124–1133 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma