Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number. ork presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.