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

Abstract

Three field mixed C 0 -continuous gradient elasticity finite element formulations contain both displacement and displacement gradients as solution variables. In this contribution a new formulation is proposed, in which the displacements are not part of the problem anymore, but only displacement gradients, leading to a reduced number of variables to be discretized. In numerical examples the proposed formulations are compared to finite strain adaptations of existing mixed three field approaches for small strains and unveil favorable computing time. For the 3D case a further reduction of variables through a penalty formulation is investigated.

Rot‐free finite elements for gradient‐enhanced formulations at finite strains

“…Here, Ψ loc (H) is a classical Neo Hooke energy function and Ψ nloc (∇H) is a quadratic gradient enhancement term (cf. [2]). On the unit square (cf.…”

“…Therefore the additional term (5) does not alter the solution of problem (3). Table 1 shows an overview of the finite element discretizations which are conforming to the Sobolev spaces of the solution variables and for which discrete inf-sup stability is proven in [2] (see also [3]). In the 3D case, an additional Lagrange multiplier µ h ∈ L 2 0 (B, IR 3 ) ∩ P 0 ensures vanishing divergence of Φ ∈ H(Div 0 ; B, (IR 3 ) 2 ).…”

“…Here u, H and Λ are the displacement, the displacement gradient and the Lagrange multiplier enforcing compatibility of the former. For a description of the Sobolev space H −1 (Div; B, (IR 3 ) 2 see [2]. For simply connected domains with connected boundary the Lagrange multiplier can be split with the decomposition [3]…”

“…Moreover, in the two-dimensional case, the Sobolev space of the rotation potential simplifies to Φ ∈ L 2 0 (B, IR). In [2] it is shown, that the variational problem corresponding to (3) with the split (4) is inf-sup stable also in the limit case of vanishing nonlocal contribution of the internal free energy density once the augmentation term…”