Mixed finite element formulations for strain-gradient elasticity problems using the FEniCS environment

• Solving mixed finite element formulation for strain gradient elasticity problems.
• Demonstrating FEniCS capabilities enabled easy implementation of the variational problems considered.
• FEniCS allows complex functionals to be quickly formulated and computed.
• Showing that the higher the degree of the discretization the better the solution agreement.

This paper presents finite element implementations for the solution of gradient elasticity problems using the FEniCS project. The FEniCS Project provides a novel tool for the automated solution of partial differential equations by the finite element method. In particular, it offers a significant flexibility with regards to modelling and numerical discretization choices. The weak form of the gradient elasticity problem is derived from the Principal of Virtual Work. An equivalent mixed-type finite element formulation for the strain gradient elasticity problem is implemented in order to avoid the use of C1 continuous elements. The complete methodology and source codes (python scripts) are provided. Numerical results are presented and compared with well-known benchmark examples, demonstrating the applicability of FEniCS for such applications.