Three‐field mixed finite element formulations for gradient elasticity at finite strains

Abstract

Gradient elasticity formulations have the advantage of avoiding geometry‐induced singularities and corresponding mesh dependent finite element solution as apparent in classical elasticity formulations. Moreover, through the gradient enrichment the modeling of a scale‐dependent constitutive behavior becomes possible. In order to remain C0 continuity, three‐field mixed formulations can be used. Since so far in the literature these only appear in the small strain framework, in this contribution formulations within the general finite strain hyperelastic setting are investigated. In addition to that, an investigation of the inf sup condition is conducted and unveils a lack of existence of a stable solution with respect to the L2‐H1‐setting of the continuous formulation independent of the constitutive model. To investigate this further, various discretizations are analyzed and tested in numerical experiments. For several approximation spaces, which at first glance seem to be natural choices, further stability issues are uncovered. For some discretizations however, numerical experiments in the finite strain setting show convergence to the correct solution despite the stability issues of the continuous formulation. This gives motivation for further investigation of this circumstance in future research. Supplementary numerical results unveil the ability to avoid singularities, which would appear with classical elasticity formulations.

A New C0‐Continuous FE‐Formulation for Finite Gradient Elasticity

“…A C 0 continuous finite element formulation for gradient elasticity at finite strains with a reduced number of solution variables has been proposed and compared to the mixed three field approach as investigated in [10]. Numerical results in 2D show appropriate convergence rates and improved computing efficiency of the proposed formulations P2 H -P1 λ and Q2 H -Q1 λ using simple standard discretizations.…”

“…[10]), the displacement u can be decoupled from the problem in the spirit of [11,12]. [10]), the displacement u can be decoupled from the problem in the spirit of [11,12].…”

“…In order to reduce the number of degrees of freedom and to avoid possible stability issues, which may occur with the three field mixed approach (c.f. [10]), the displacement u can be decoupled from the problem in the spirit of [11,12]. The reformulation of (2.1) now reads…”

“…For the dimensions d ∈ {2, 3} (2D and 3D), the continuous function spaces of the solution variables are defined as follows [10]: Through the introduction of H only first order derivatives appear in the corresponding weak formulation and thus, the continuity requirement is relaxed to C 0 .…”

“…In what follows for discrete functions u h and H h , the second order Lagrange approximation will be considered while for λ h first order Lagrange approximation is chosen (c.f. [8,10]). The corresponding finite elements will be denoted as P2 u -P2 H -P1 λ and Q2 u -Q2 H -Q1 λ for simplical and quadrilateral or hexahedral elements respectively.…”