2D Elastoplastic Problems Research Articles

Exact satisfaction of boundary and interface conditions in nodal-integration-based finite element methods

In spite of some drawbacks, finite element methods based on nodal rather than Gaussian integration present major advantages, especially in the context of elastoplastic or elastoviscoplastic problems—notably the elimination of locking problems due the plastic or viscoplastic incompressibility condition, and the reduction of computation and storage requirements pertaining to internal variables. This paper investigates another potential advantage of such methods, namely the possibility to account exactly—instead of approximately like with Gaussian integration—for conditions of prescribed traction on external surfaces, and continuity of the traction-vector across internal interfaces separating different materials. The technique proposed is somewhat similar to that classically used to satisfy plane stress conditions in 2D elastoplastic problems: it consists, when using the constitutive law to evaluate the stresses from the strains, in adjusting the out-of-plane components of the strain, so as to enforce either identity of the traction-vector and its prescribed value on external surfaces, or identity of the traction-vectors on both sides of internal interfaces. The examples provided for both traction-free boundaries and interfaces between materials evidence the efficiency of the technique.

PIES for 2D elastoplastic problems with singular plastic strain fields

The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).

On the Stress Fluctuation in the Smoothed Finite Element Method for 2D Elastoplastic Problems

In this study, the unusual stress fluctuation in the smoothed finite element method (SFEM) for 2D elastoplastic problems is investigated. The governing equation of the 2D elastoplastic problem based on SFEM is presented. The mysterious stress fluctuations from different SFEM approaches are explored. It is demonstrated that conventional finite element method might experience the same symptom. It is found that the stress noises only occur for cases consisting of elements with linear shape functions. Three strategies, including use of higher-order elements, modification of interpolation techniques, and evenly distributed mesh division are proposed to prevent mysterious stress fluctuations. All of them could considerably minimize the strange stress noises.

An object-oriented approach to dual reciprocity boundary element method applied to 2D elastoplastic problems

PurposeThe purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method (DRM) but using object-oriented programming (OOP). As the BEM formulation generates a domain integral due to the inelastic stresses, the DRM is employed in a modified form using polyharmonic spline approximating functions with polynomial augmentation. These approximating functions produced accurate results in BEM applications for a range of problems tested, and have been shown to converge linearly as the order of the function increases.Design/methodology/approachA programming class named DRMOOP, written in C++ language and based on OOP, was developed in this research. With such programming, general matrix equations can be easily established and applied to different inelastic problems. A vector that accounts for the influence of the inelastic strains on the displacements and boundary forces is obtained.FindingsThe C++ DRMOOP class has been implemented and tested with the BEM formulation applied to classical elastoplastic problem and the results are reported at the end of the paper.Originality/valueAn object-oriented technology and the C++ DRMOOP class applied to elastoplastic problems.

Modification of the boundary integral equation for the Navier–Lamé equations in 2D elastoplastic problems and its numerical solving

SummaryThe paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2D elastoplastic problems. The modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements. This was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation. For approximation of plastic strains the Lagrange polynomials with various number and arrangement of interpolation nodes were used. Reliability of the modification was verified on examples with analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.

Investigation of RPIM Shape Parameter Effects on the Solution Accuracy of 2D Elastoplastic Problems

The effects of radial point interpolation method (RPIM) shape parameters on the solution accuracy of 2D elastoplastic problems are investigated. The RPIM algorithm is adapted for the solution of 2D elastoplastic problems using the multi-quadric radial basis function. The convergence rates after yielding are investigated for perfectly plastic and hardening cases with various shape parameters. Both regular and irregular node distributions are used by considering the standard Gauss integration and nodal integration schemes. It is shown that the solutions after yielding can be improved using appropriate shape parameters. The propagation of plastic region and solution time of each case are presented against load increments.