On the spatial and material motion problems in nonlinear electro-elastostatics with consideration of free space

Abstract

In this work the formulation of spatial and material motion problems in nonlinear electro-elastostatics is considered using an energy approach, which takes into account the contribution of the free space surrounding a nonlinearly polarized body undergoing large deformation. The free space can have a huge impact on the electric field and on the deformation field inside a body made of materials with low electric permittivity such as the so-called electronic electroactive polymers (EEAPs). The contribution of the free space can be taken into account by using the electric flux and the Maxwell's traction acting on the boundary of the body. These two quantities can be expressed in terms of a stored energy density function. By using a stored energy density function for both the material body and the (finite or infinite) free space, the governing equations of both spatial and material motion problems are derived by considering the change of energy with respect to a change in the spatial or material configuration. In the spatial motion problem, well-known definitions for electric and mechanical quantities are derived. In the material motion problem, in addition to the derivation of configurational forces, this approach reveals the formulas for the part of energy that is released from the system material body, applied forces in response to a change in the material configuration, which are particularly useful in the study of defects such as crack propagation. The same approach can be used in the case of nonlinear electro-thermo-mechanical coupling and constitutes the direction for future works.

Variable kinematics models and finite elements for nonlinear analysis of multilayered smart plates

“…(5), (7) and (8) hold, in the absence of body mechanical forces and electric charge, the problem governing equations are obtained in terms of the primary variables u; U and W by the following variational statement [28,27] …”

“…The electric state is accounted by the material electric field vector E and the conjugate material electric displacements D [26,27], which are also partitioned into their in-plane and out-of-plane components ð2aÞ ð2bÞ Analogously, the magnetic state is described by the material magnetic field vector H and the material magnetic induction vector B, partitioned as…”

On weak formulations and their second variation in nonlinear electroelasticity

“…Different works followed and we refer to Maugin (1977), the monographs by Eringen and Maugin (1990) and Maugin (1988) for further details. With respect to variational formulations we mention the work by Yang and Batra (1995).…”

Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

“…As already mentioned, in order to account for the strong contraction induced by the electromechanical coupling, the behavior of active bio-materials must be described in terms of finite deformations, following the lines of relevant recent works, see, e.g., [25,52,62,70,71,75,76]. The standard approach for modeling bio-active tissues is based on a passive response expressed in terms of hyperelastic weakly compressible or incompressible materials, combined to an active response (or active stress) formulated in an independent way, in several cases as a phenomenological description of some inelastic action.…”