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 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.
Highlights
Stress gradient elasticity is a generalised continuum theory that has been proposed very recently as a counterpart of Mindlin’s well-known strain gradient elasticity model [14, 25, 26]
Stress gradient elasticity should neither be confused with the Aifantis gradient elasticity model which is a special case of isotropic strain gradient elasticity [13, 29], nor with models based on Laplacians of stress and strain which have been recently recognised as special cases of micromorphic elasticity [7, 10, 16]
The strain gradient model is solely based on the classic displacement degrees of freedom
Summary
Stress gradient elasticity is a generalised continuum theory that has been proposed very recently as a counterpart of Mindlin’s well-known strain gradient elasticity model [14, 25, 26]. Applications of stress gradient elasticity have been rather limited up to now They include prediction of size effects in bending with a comparison with strain gradient and micromorphic approaches [28], and particle size effects in composite materials [33]. The latter reference contains extensions of the Eshelby and Hashin-Shtrikman homogenisation approaches to heterogeneous stress gradient media in a simplified case of isotropic elasticity. The observed size effects with respect to structure size and the intrinsic length scale of the model will be discussed in detail This contribution is organised as follows: after a brief recapitulation of the fundamentals of the stress gradient theory, a finite element formulation is discussed in detail in Sect.
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