Some C-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems

• Published in: International journal of solids and structures 2008-06, Vol.45 (11-12), p.3255-3281
• Main Authors: Markolefas, S I, Tsouvalas, D A, Tsamasphyros, G I
• Format: Article
• Language: English
• ISSN: 0020-7683
• DOI: 10.1016/j.ijsolstr.2008.01.021

The goal of this work is a systematic presentation of some classes of mixed weak formulations, for general multi-dimensional dipolar gradient elasticity (fourth order) boundary value problems. The displacement field main variable is accompanied by the double stress tensor and the Cauchy stress tensor (case 1 or mu - tau - u formulation), the double stress tensor alone (case 2 or mu - u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or mu - tau - kappa - epsilon - u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u - theta - gamma formulation). In all formulations, the respective essential conditions are built in the structure of the solution spaces. For cases 1, 2 and 4, one-dimensional analogues are presented for the purpose of numerical comparison. Moreover, the standard Galerkin formulation is depicted. It is noted that the standard Galerkin weak form demands C1-continuous conforming basis functions. On the other hand, up to first order derivatives appear in the bilinear forms of the current mixed formulations. Hence, standard C0-continuous conforming basis functions may be employed in the finite element approximations. The main purpose of this work is to provide a reference base for future numerical applications of this type of mixed methods. In all cases, the associated quadratic energy functionals are formed for the purpose of completeness.