Formulation of plane finite element models for gradient elasticity and plasticity analyses

Abstract

Computational analysis of many gradient-enhanced elasticity/plasticity theories requires the interpolation of the displacement/plasticity variable to be first-order continuous, i.e. C1. However, the construction of C1 finite element shape functions is not trivial. In this thesis, some advanced finite element formulations for thin plate analysis are revisited and generalized to devise finite element models for gradient-enhanced elasticity/plasticity. These include the discrete Kirchhoff method (DK), hybrid stress method (HS), and the relax hybrid method (RH). Besides, the finite cell method (FCM) is implemented using the C1 Hermite element for gradient-enhanced elasticity and plasticity analyses. In the generalized DK method for strain-gradient elasticity, the displacement and displacement-gradient are interpolated separately. The element stiffness is computed directly from the strain and the second-order displacement-derivatives derived from the interpolated displacement and displacement-gradient, respectively. Three-node triangular and four-node quadrilateral elements are devised. For nearly incompressible analysis, the accuracy of the generalized DK elements is improved using the displacement-pressure mixed formulation. In the HS finite element method for strain-gradient elasticity, the two-field Hellinger-Reissner functional is employed. By employing equilibrating stress and double-stress modes, the C1 displacement is only required to be defined only on the element boundary but not the element domain. Three-node triangular and four-node quadrilateral element models are devised. They satisfy the constant double-stress patch test and converge in the numerical examples. Moreover, they are free from dilatational locking. In the RH formulation for strain-gradient elasticity, a generalized two-field Hellinger-Reissner functional is employed. The displacement is interpolated with zeroth-order continuous shape functions. The assumed constant double-stress is employed to enforce the continuity of the displacement normal-gradient. By employing the basis of the displacement-derived double-stress as that of the assumed double-stress, the derivation of element stiffness can be simplified. Three-node triangular and four-node quadrilateral elements are devised. For the FCM for strain-gradient elasticity, the rectangular C1 Hermite element is taken to be the basic computational cell. In the FCM, the problem is solved in a larger extended domain that can be discretized with rectangular elements. The influence of the extended part is eliminated by multiplying the material constants with a small scaling factor. In this work, an integration scheme that divides the cut-cells into integration sub-zones is employed. To enrich the local trial solution space, a discretization scheme that replaces the cut-cells with smaller rectangular cells and cut-cells. Lastly, the interpolation methods of the quadrilateral DK, quadrilateral RH and the BFS Hermite elements are employed to formulate finite elements for gradient-plasticity analysis. Three finite element models are devised, and they yield close predictions in numerical examples. In summary, several advanced finite element formulations for thin plate analyses are generalized to strain-gradient elasticity. The devised element models usually yield more accurate predictions than the existing mixed/penalty-type finite element methods. Furthermore, the application of the formulations to gradient-plasticity analysis is also explored. (464 words)