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A finite element implementation of the stress gradient theory

Kaiser, Tobias ; Forest, Samuel and Menzel, Andreas LU (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
; and
organization
publishing date
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}},
}