Eulerian formulation of the constitutive relation for an electro-magneto-elastic material class

Abstract

A novel class of electro-magneto-elastic (EME) materials comprise electro-active and magnetoactive particles in the polymer matrix that change their elastic behavior with an applied electromagnetic field. The material response for such a material class is usually formulated in terms of Lagrangian strain tensor along with Lagrangian electromagnetic field vectors as "pushed forward" to the current configuration. This letter article presents a novel formulation of an electro-magnetoelasticity in terms of an Eulerian strain tensor and Eulerian electromagnetic field vectors referring to the current configuration. Such an Eulerian formulation is often favorable from both theoretical and computational standpoints, which avoids the "pushed forward" operation to get the current configuration. Additionally, an exercise to deduce the constitutive relation for an EME material class available in the existing literature from the newly proposed relation is also illustrated.From experience, we know that two bodies of the same 'shape' and dimensions, with the same distribution of external load but made of different materials, will behave differently [1,2,3]. Connecting the 'deformation' with the stresses became necessary in such conditions [4,5]. Generally, such connections are the simplified mathematical models relating the 'deformation' with the 'external loads', known as the constitutive relations for any material class [6,7,8]. Such relations primarily aim to represent approximately the material behavior without intending to model this behavior in all the possible external conditions. In this regard, to track the changes in elastic behavior of a smart elastomer responding to an applied electromagnetic field, another set of constitutive relations must be derived from the theory of electro-magneto-elasticity [9,10]. In such relations, the material response is usually formulated using two commonly known Lagrangean and Eulerian strain tensors. Lagrangean and Eulerian strain tensors are generally related through the "pushed forward" and "pushed backward" operations. The "pushed forward" operation is a process of taking a Lagrangian quantity and writing its Eularian analog and vice-versa for reverse. The necessity of using Eulerian-based relations is often favorable from both theoretical and computational standpoints, which avoid the "pushed forward" operation to get the current configuration. In the reference configuration of an EME material class, the electro-active and magneto-active particles are not aligned. Rather, they are random. So, upon applying an electric field, magnetic field, or both together, the individual particles are now aligned in the current configuration, and the defined stress tensor will have a physical sense. But, when we convert all the variables to the reference configuration, that configuration becomes a pseudo configuration, not the actual one. So, to avoid such confusion, we propose an Eulerian approach to get the direct physical sense and the correct measure of the stresses, including all the kinemati...