A finite element implementation of the stress gradient theory

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.

Influence of Topology and Porosity on Size Effects in Stripes of Cellular Material with Honeycomb Structure under Shear, Tension and Bending

“…The constants of integration in ( 13) have been determined by the trivial boundary condition σ 11 (±H/2) = 0. A plot of the stress distribution in Figure 22 exhibits similarities with the torsion solution in [24]. The second equation (12) 2 would now allow to determine the lateral contraction u lat (x 2 ).…”

A benchmark strain gradient elasticity solution in two-dimensions for verifying computational approaches by means of the finite element method

“…Generalized mechanics has been widely investigated in the literature. It has been implemented for problems of elasticity [9][10][11][12][13]; plasticity [14][15][16][17][18][19]; damage modeling [20][21][22][23][24][25]; modeling metamaterials [26][27][28] such as pantographic structures [29][30][31], network materials [32], viscoelastic truss structures [33], bipantographic structures [34], second gradient fluids [35]; gradient-enhanced homogenization [36][37][38][39]; micropolar continua [40]; fracture mechanics [41]; biomechanics [42][43][44]; and anisotropic systems [45]. Parameter determination of generalized mechanics models has been studied for static and dynamic regimes in Shekarchizadeh et al [46,47], respectively.…”

On the torsion of isotropic elastoplastic Cosserat circular cylinders

“…The torsion problem is a particular case of the Saint-Venant problem and is the subject of the present work. Note that the torsion problem has been solved for several other generalized continua like strain gradient elastic media [Lazopoulos and Lazopoulos, 2012;Iesan, 2013;Beheshti, 2018], and recently elastic stress gradient media [Kaiser et al, 2021]. The torsion of Cosserat bars is of physical relevance for the identification of Cosserat elasticity moduli and associated internal length scales [Gauthier and Jashman, 1975].…”