A multiplicative approach for nonlinear electro-elasticity

Abstract The recent interest in dielectric elastomers has given rise to a pressing need for predictive non-linear electromechanical coupling models. Since elastomers behave elastically and can sustain large deformations, the constitutive laws are naturally based on the formulation of adequate free energy functions. Due to the coupling, such functions include terms which combine the strain tensor and the electric field. In contrast to existing frameworks, this paper proposes to establish the electromechanical coupling by the multiplicative split of the deformation gradient into a part related to the elastic behavior of the material and further one which describes the deformation induced by the electric field. Already available and well tested functions of elastic free energy functions can be immediately deployed without any modifications provided the argument of the function is the strain tensor alone which in turn is defined by the elastic part of the deformation gradient only. An appropriate constitutive relation is formulated for the electrically induced part of the deformation gradient. The paper discusses in depth such a formulation. The approach is elegant, straightforward and above all, provides clear physical insight. The paper presents also a numerical formulation of the theoretical framework based on a meshfree method. Various numerical examples of highly non-linear coupled deformations demonstrate the potential and strength of the theory.