A finite element method for local second gradient model using Lagrange multipliers

Second gradient terms are well known to be a remedy for regularizing the solutions of problems involving models used beyond localization. Usually second gradient models are non local. Recently we proposed local second gradient models. Analytically it is possible to get small strain solutions in the one dimensional case. Numerically, it is easy to develop a one dimensional Hermitian Finite Element Method assuming small strain assumption as well as large strain one. However this cannot be generalized for 2D or 3D problems. In this paper, the proposed local second gradient models are seen as a particular case of microstructured media which obeys a kinematic constraint. It is then possible to build up a large (as well as a small) strain Finite Element Method useful for 1D as well as for 2D or 3D problems, using Lagrange multipliers. This Finite element method is based on a full Newton-Raphson method and the well known notion of consistent tangent stiffness matrix. One dimensional examples are given which show the power of the proposed method. This way has been proved to be efficient for the simple case of 1D problem. Generalization for 2D problem is a work in progress and seems to give results as good as for the one dimensional case.