Some C0-continuous mixed formulations for general dipolar linear gradient elasticity boundary value problems and the associated energy theorems

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 μ − τ − u formulation), the double stress tensor alone (case 2 or μ − u formulation), the double stress, the Cauchy stress, the displacement second gradient and the standard strain field (case 3 or μ − τ − κ − ε − u formulation) and the displacement first gradient, along with the equilibrium stress (case 4 or u − θ − γ 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.