A finite element implementation of the stress gradient theory
(2021) In Meccanica 56(5). p.1109-1128- Abstract
In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of... (More)
In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.
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- author
- Kaiser, Tobias ; Forest, Samuel and Menzel, Andreas LU
- organization
- publishing date
- 2021
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Analytical solutions, Finite elements, Generalised continuum, Strain gradient elasticity, Stress gradient elasticity, Stress gradient theory
- in
- Meccanica
- volume
- 56
- issue
- 5
- pages
- 1109 - 1128
- publisher
- Springer
- external identifiers
-
- scopus:85102083805
- ISSN
- 0025-6455
- DOI
- 10.1007/s11012-020-01266-3
- language
- English
- LU publication?
- yes
- id
- 8806e92c-180f-4c81-9caa-dba5eaa8247a
- date added to LUP
- 2021-03-19 08:37:34
- date last changed
- 2022-04-27 00:53:52
@article{8806e92c-180f-4c81-9caa-dba5eaa8247a, abstract = {{<p>In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.</p>}}, author = {{Kaiser, Tobias and Forest, Samuel and Menzel, Andreas}}, issn = {{0025-6455}}, keywords = {{Analytical solutions; Finite elements; Generalised continuum; Strain gradient elasticity; Stress gradient elasticity; Stress gradient theory}}, language = {{eng}}, number = {{5}}, pages = {{1109--1128}}, publisher = {{Springer}}, series = {{Meccanica}}, title = {{A finite element implementation of the stress gradient theory}}, url = {{http://dx.doi.org/10.1007/s11012-020-01266-3}}, doi = {{10.1007/s11012-020-01266-3}}, volume = {{56}}, year = {{2021}}, }