On the Analysis of an EFG Method Under Large Deformations and Volumetric Locking

Abstract

This work investigates a modified element-free Galerkin (MEFG) method when applied to large deformation processes. The proposed EFG method enables the direct imposition of the essential boundary conditions, as a result of the kronecker delta property of the special shape functions, constructed in the neighborhood of the essential boundary. The plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and a plastic part and considers a J 2 elasto-plastic constitutive relation that accounts for a nonlinear isotropic hardening. The constitutive model is written in terms of the rotated Kirchhoff stress and of the conjugate logarithmic strain measure. A total Lagrangian formulation is considered in order to improve the computational performance of the proposed algorithm. Here, aspects related to the volumetric locking are numerically investigated and an F-bar approach is considered. Some numerical results are presented, under axisymmetric and plane strain assumption, in order to attest the performance of the proposed method.

Elastoplastic Large Deformation Using Meshless Integral Method

“…In mesh- ). A number of techniques for enforcing essential boundary conditions have been developed, including the collocation methods [33], Lagrange multiplier method [34], penalty method [35,36], Nitsche's method [37], coupled meshless-finite element method [38], and other methods [39][40][41][42][43]. The advantages and disadvantages of a variety of methods for enforcing essential boundary conditions have been discussed briefly in [1].…”

“…Generally, there are two types of discretization in meshless methods: 1) Galerkin based method over the global domain, which is used in EFG [33,42,47,48], clouds [41], RKPM [49][50][51][52]; and 2) local weak form over multiple local domains, which is employed in [31,[53][54][55]. For Galerkin based method over the global domain, the n system equations are obtained by applying the weak form over the global domain once, hence all equations must hold simultaneously in order to maintain consistency of the weak form.…”

“…Belytschko et al [8] proposed a 3D Element-Free Galerkin (EFG) method intended for dynamic problems with geometric and material nonlinearities solved with explicit time integration. Rossi and Alves [9] applied a modified Element-Free Galerkin method to large deformation processes. The proposed EFG method enables direct enforcement of essential boundary conditions, and the plasticity model assumes a multiplicative decomposition of the deformation gradient into an elastic and plastic part and allows for nonlinear isotropic hardening.…”

“…A Galerkin SPH method was utilized by Wang to solve large deformation fracture problems [23]. Reproducing kernel particle methods were used to analyze large deformation problems in [24][25][26]. In [27], Li et al used an element-free Galerkin method to solve die forging problems.…”

Analysis of Axisymmetric Problems by Element-Free Galerkin Method

“…Since the development of Element-Free Galerkin (EFG) method by T. Belytschko et al [1], rather similarity of the method to the traditional FEM together with its higher accuracy, has made it the most well-known mesh-free method amid others. In the past, many aspects of this method have been addressed; and it has been used in solving many different problems of solid mechanics; such as crack propagation [2], [3], elasto plasticity [4], [5], large deformation [6] and lately in geomechanics [7], [8]. Three-dimensional problems have also been generally featured in the literature [2], [4].…”