Rot‐free mixed finite elements for gradient elasticity at finite strains

Abstract

Through enrichment of the elastic potential by the second-order gradient of deformation, gradient elasticity formulations are capable of taking nonlocal effects into account. Moreover, geometry-induced singularities, which may appear when using classical elasticity formulations, disappear due to the higher regularity of the solution. In this contribution, a mixed finite element discretization for finite strain gradient elasticity is investigated, in which instead of the displacements, the first-order gradient of the displacements is the solution variable. Thus, the C 1 continuity condition of displacement-based finite elements for gradient elasticity is relaxed to C 0. Contrary to existing mixed approaches, the proposed approach incorporates a rot-free constraint, through which the displacements are decoupled from the problem. This has the advantage of a reduction of the number of solution variables. Furthermore, the fulfillment of mathematical stability conditions is shown for the corresponding small strain setting. Numerical examples verify convergence in two and three dimensions and reveal a reduced computing cost compared to competitive formulations. Additionally, the gradient elasticity features of avoiding singularities and modeling size effects are demonstrated.

A procedure for the experimental identification of the strain gradient characteristic length

“…Although the use of strain gradient models is not new [15,12,20], and special Ąnite element schemes have been developed [28,29,30], one of the main challenges associated with these models is the difficulty in determining the constitutive coefficients [19]. Strain gradient models [23,24,14], incorporate additional terms in the constitutive equations which depend on the materialŠs microstructure and the speciĄc deformation modes that occur during loading [4], which can make them difficult to be determined experimentally [2].…”