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.
Highlights
The internal elastic free energy density of classical hyperelastic formulations is a function of the first-order gradient of deformation
Gradient elasticity formulations have the advantage of avoiding geometry-induced singularities and corresponding mesh dependent finite element solution as apparent in classical elasticity formulations
An investigation of meshless discretization methods for the gradient elasticity formulation can be found in Askes and Aifantis,[5,23] whereas in Papanicolopulos et al.[24] a C1-continuous finite element with a preprocessing smoothing based on a least squared minimization is presented
Summary
The internal elastic free energy density of classical hyperelastic formulations is a function of the first-order gradient of deformation. An investigation of meshless discretization methods for the gradient elasticity formulation can be found in Askes and Aifantis,[5,23] whereas in Papanicolopulos et al.[24] a C1-continuous finite element with a preprocessing smoothing based on a least squared minimization is presented The drawback of the latter one is the restriction to structured meshes only. Based on the idea of Ru et al.,[25] in which the fourth-order gradient elasticity problem of Aifantis[15] is split into two second order problems, a C0-continuous corresponding finite element formulation is possible.[26] While the latter is restricted to the special formulation of Aifantis,[15] in Shu et al.[27] and Zybell[28] a mixed approach is investigated, which enables C0-continuous discretizations in the framework of the more general Mindlin-Toupin gradient elasticity theory.
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