Simplifying the mixed finite-element formulation for strain-gradient problems

Research output: Contribution to conferenceAbstractpeer-review

Abstract / Description of output

he finite-element implementation of strain-gradient theories, frequently proposed for modelling the micromechanical behaviour of materials, is not straightforward. In a displacement-only finite-element formulation, the presence of strain gradients (i.e. second derivatives of the displacements) leads to the requirement for elements with C1 continuous interpolation. While such elements have been successfully proposed and used [1,2], alternative mixed formulations have been sought early on to allow the use of more common element shape functions. These mixed formulations are based on the interpolation of two different fields, displacement and some kind of displacement gradient, with the relation between the two fields enforced using either Lagrange multipliers [3,4,5] or penalty methods [5,6].
In this work, we use a recently developed framework [7] to critically review earlier and recent mixed elements for strain-gradient theories that have been proposed in the literature. We then consider systematically ways in which the computational cost can be decreased while retaining accuracy of the results. An interesting concept here is the determination of the minimal content requirements for the displacement gradient field. We then present a new approach to mixed elements for strain-gradient theories that avoids the need for either Lagrange multipliers or penalty parameters. We provide a comparative evaluation of the proposed approach, discussing issues with the application of boundary conditions and how the solution process can be adapted to address these.

References
[1] A. Zervos, P. Papanastasiou, I. Vardoulakis (2001): A finite element displacement formulation for gradient elastoplasticity. Int. J. Numer. Methods Eng. 50(6), 1369–1388.
[2] S.-A. Papanicolopulos, A. Zervos, I. Vardoulakis (2009): A three dimensional C1 finite element for gradient elasticity. Int. J. Numer. Methods Eng. 77(10), 1396–1415.
[3] J. Y. Shu, W. E. King, N. A. Fleck (1999): Finite elements for materials with strain gradient effects. Int. J. Numer. Methods Eng. 44(3), 373–391.
[4] T. Matsushima, R. Chambon, D. Caillerie (2002): Large strain finite element analysis of a local second gradient model: application to localization. Int. J. Numer. Methods Eng. 54(4), 499–521.
[5] V. G. Kouznetsova, M. G. D. Geers, W. A. M. Brekelmans (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Eng. 193(48), 5525–5550.
[6] A. Zervos (2008): Finite elements for Elasticity with Microstructure and Gradient Elasticity. Int. J. Numer. Methods Eng. 73(4), 564–595.
[7] S.-A. Papanicolopulos, F. Gulib, A. Marinelli (2019): A novel efficient mixed formulation for strain-gradient models. Int. J. Numer. Methods Eng. 117(80, 926–937.
Original languageEnglish
Publication statusPublished - Apr 2022
EventUK Association for Computational Mechanics Annual Conference - Nottingham, United Kingdom
Duration: 20 Apr 202222 Apr 2022
https://www.ukacm2022.ukacm.org/

Conference

ConferenceUK Association for Computational Mechanics Annual Conference
Abbreviated titleUKACM 2022
Country/TerritoryUnited Kingdom
CityNottingham
Period20/04/2222/04/22
Internet address

Fingerprint

Dive into the research topics of 'Simplifying the mixed finite-element formulation for strain-gradient problems'. Together they form a unique fingerprint.

Cite this