Optimization method of hybrid element stress function for strain gradient theory based on Hellinger-Reissner principle
基于Hellinger-Reissner变分原理的应变梯度杂交元设计

Recent experiments have shown that materials will display strong scale effect when the scale of non-uniform plastic deformation field associated their intrinsic length scale is on the order of microns. In order to explain such scale effect phenomena, Fleck and Hutchinson developed a couple stress theory of strain gradient plasticity based on the reduced couple stress theory, which incorporates the rotation gradient of deformation into constitutive model, and introduces a material characteristic length parameter related to the rotation gradient. Theoretical predictions agree well with the micro-torsion and micro-bending experiments. In the finite element implementation of Fleck-Hutchinson couple stress plasticity, the higher order nature of theory requires that both the displacement and its first-order derivatives to be continuous across the adjacent elements' boundaries. Noticed that the micro-rotation *********, an independent kinematic quantity with no direct dependence on displacement u, is introduced in the general couple stress theory. This enables the C0-continuous element to be developed based on the general couple stress theory. Fitting within the framework of general couple stress theory, the energy consistency condition of the discrete finite element system for couple stress strain gradient theory is derived by introduction of incompatible displacement trial functions. Furthermore, the optimization condition of stress trial functions for hybrid element of strain gradient theory is constructed based the energy consistency condition. A 4-node C0 kind hybrid element is designed in terms of the optimization condition. Numerical tests show that the scale effects can be reflected with the element designed in the paper and reliable results is delivered both for compressible and incompressible materials.