Optimization method of hybrid element stress function for strain gradient theory based on Hellinger-Reissner principle

Recent experiments have shown that materials will displaystrong scale effect when the scale of non-uniform plastic deformation fieldassociated their intrinsic length scale is on the order of microns. In orderto explain such scale effect phenomena, Fleck and Hutchinson developed acouple stress theory of strain gradient plasticity based on the reducedcouple stress theory, which incorporates the rotation gradient ofdeformation into constitutive model, and introduces a materialcharacteristic length parameter related to the rotation gradient.Theoretical predictions agree well with the micro-torsion and micro-bendingexperiments.In the finite element implementation of Fleck-Hutchinson couple stressplasticity, the higher order nature of theory requires that both thedisplacement and its first-order derivatives to be continuous across theadjacent elements' boundaries. Noticed that the micro-rotation ω, an independent kinematic quantity with no direct dependence ondisplacement u, is introduced in the general couple stresstheory. This enables the C0-continuous element to be developed based onthe general couple stress theory. Fitting within the framework of generalcouple stress theory, the energy consistency condition of the discretefinite element system for couple stress strain gradient theory is derived byintroduction of incompatible displacement trial functions. Furthermore, theoptimization condition of stress trial functions for hybrid element ofstrain gradient theory is constructed based the energy consistencycondition. A 4-node C0 kind hybrid element is designed in terms of theoptimization condition. Numerical tests show that the scale effects can bereflected with the element designed in the paper and reliable results isdelivered both for compressible and incompressible materials.