Abstract: The scope of the present work is to present Hu – Washizu & Hellinger-Reissner type energy functionals for various mixed formulations in the framework of the multidimensional linear Strain Gradient Theory developed by Mindlin & co-workers.
The main characteristic of the linear Strain Gradient theory is the consideration of the existence of microstructure in the material, which in turn gives rise to a great amount of new independent kinematic variables. These new variables lead to complicated forms of main boundary value problem. For example, it is easy to verify that the equilibrium equations are of the 4^th order with respect to the displacement field. Another, hindrance to the direct application of these theories is the fact that the complicated boundary conditions that appear in this framework are somewhat difficult to interpret physically.
With the above in mind, various mixed formulations (of the Ciarlet-Raviart type) have been developed in order to incorporate combinations of variables that include, but do not limit to, the classic stress (ô) & strain (å), displacement field (u), double stress (ì), relative stress (ã) and 1^st (è) & 2^nd (k) gradient of the displacement field. For reasons of completeness the following mixed formulations are proposed 1) ì-ô-u 2) ì-u 3) ì-ô-k-å-u 4) è-u-ã.
It should be noted that in this case a direct extension to the classic Hu – Washizu & Hellinger-Reissner energy functionals (as they appear in the standard elasticity), cannot be made due to the great plethora of viable combinations among the classic and microstructural kinematic variables. However, as it is shown in the present work, similar energy functionals can be stated due to the fact that all of the above stated mixed formulations originate from the form  and that their solution minimizes the functional .
Another implication of the current work is that it can be seen as a broad reference base for future numerical implementations of these theories by means of, for instance, various weighted residual methods such as the Standard Galerkin method.

Brief Biography of the Speaker:
George É. Tsamasphyros is a Professor of Computational Mechanics at the Faculty of Applied Mathematical and Physical Sciences of the National Technical University of Athens (NTUA), Greece. He received the B.S. in Civil Engineering from the NTUA and his M.S in Applied Mathematics / Mechanics of Solids as well as the PhD degree (Doctor of Sciences) from the University of Paris VI.
He has supervised 14 PhD candidates and teaches several BA and MA courses. Dr G. Tsamasphyros is the author of over of 100 published papers in referee journals, 80 papers published in International Conferences and 12 books, all of them are used as textbooks in NTUA and other Greek Universities. He has been chairman of the Department of Mechanics, vice-dean of NTUA, Secretary of the Ministry of Education-Responsible of matters of the European Union.
His research interests are Computational Mechanics, (covering a wide spectrum of it: Finite Element Method, Integral Equations, Boundary Element Method, Integral Transformations, Finite Differences, and Finite Volumes). Special emphasis is given to finding new methods for calculating the error and the convergence of these methods. The boundary value problems which are confronted concern elasticity problems, fracture mechanics, as well as problems of gradient elasticity (materials with microstructure) and coupled fields – piezoelectric materials, composite materials, structure repair, wear and fatigue of materials, as well as biomechanics issues.
Another aspect of his research concerns the use of contemporary methods of sensing and measurement (optical - Bragg Gratings – Magnetic and Piezoelectric Sensors) for monitoring of structures emphasizing on the control of structural integrity by using neural networks.