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ZHANG Ji-fa, ZHANG Wen-pu, ZHENG Yao. A meshfree method and its applications to elasto-plastic problems[J]. Journal of Zhejiang University Science A, 2005, 6(2): 148~154.

@article{title="A meshfree method and its applications to elasto-plastic problems",

author="ZHANG Ji-fa, ZHANG Wen-pu, ZHENG Yao",

journal="Journal of Zhejiang University Science A",

volume="6",

number="2",

pages="148~154",

year="2005",

publisher="Zhejiang University Press & Springer",

doi="10.1631/jzus.2005.A0148"

}

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%T A meshfree method and its applications to elasto-plastic problems

%A ZHANG Ji-fa

%A ZHANG Wen-pu

%A ZHENG Yao

%J Journal of Zhejiang University SCIENCE A

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%I Zhejiang University Press & Springer

%DOI 10.1631/jzus.2005.A0148

TY - JOUR

T1 - A meshfree method and its applications to elasto-plastic problems

A1 - ZHANG Ji-fa

A1 - ZHANG Wen-pu

A1 - ZHENG Yao

J0 - Journal of Zhejiang University Science A

VL - 6

IS - 2

SP - 148

EP - 154

%@ 1673-565X

Y1 - 2005

PB - Zhejiang University Press & Springer

ER -

DOI - 10.1631/jzus.2005.A0148

**Abstract: **Standard finite element approaches are still ineffective in handling extreme material deformation, such as cases of large deformations and moving discontinuities due to severe mesh distortion. Among meshfree methods developed to overcome the ineffectiveness, Reproducing Kernel Particle Method (RKPM) has demonstrated its great suitability for structural analysis. This paper presents applications of RKPM in elasto-plastic problems after a review of meshfree methods and an introduction to RKPM. A slope stability problem in geotechnical engineering is analyzed as an illustrative case. The corresponding numerical simulations are carried out on an SGI Onyx3900 supercomputer. Comparison of the RKPM and the FEM under identical conditions showed that the RKPM is more suitable for problems where there exists extremely large strain such as in the case of slope sliding.

**
**

. INTRODUCTION

All methods in this family, such as Smooth Particle Hydrodynamics (SPH) (Lucy,

Among these methods, the EFG and the RKPM have been demonstrated as most suitable for structural analysis. Taking into account that the RKPM provides a general formulation for the construction of shape functions for meshfree methods, with specific discretization of the reproduced equation, so that the SPH and the EFG methods can be recovered. We present only the implementation of RKPM in this paper. Chen et al.(

In the present paper, after a brief introduction to the RKPM, we present a meshfree discretization method for elasto-plastic problems with the Drucker-Prager and Mohr-Coulomb models that had been tested and proved suitable for geotechnical materials. A slope stability problem in geotechnical engineering is chosen as a sample case to demonstrate the advantages of the meshfree methods.

. REVIEW OF THE RKPM

in which

and

and

The extension of Φ

In this construction, the supports of kernel functions are rectangular and hexahedral in geometries, each with a center

The correction function for the multi-dimensional case is expressed as

and Δ

. RKPM FOR ELASTO-PLASTIC PROBLEMS

Recall that

and

are nodal values.

By using the expressions obtained above, we can discretize the weak forms of the elasto-plastic problems. As we known well, the equilibrium equation of the small strain elasto-plastic problem can be written as

Its weak form based on virtual displacement principle can be written as

with the constitutive relations

Similar to those in finite element methods, the following expressions must hold theoretically

in which

in which

In addition, though Eq.(

for evaluating the material tangent stiffness matrix.

At any stage, the incremental form of Eq.(

Based on this equation, the numerical procedure, such as Newton-Raphson Method and Tangential Stiffness Method (sometimes named as Generalized Newton-Raphson Method), can be directly applied.

. CASE STUDY

A quadratic stored energy function

Let the strain-like vector of plastic internal variables

A Drucker-Prager yield function takes the form

with derivatives

The value

A plastic potential function

with derivatives

With the plastic potential function

Note that

which is analogous to the dilatancy factor used by Rudnicki (

There are many kernel functions proposed. Here we employ a cubic spline function as the kernel function of Eq.(

As an illustrative case, we study a plane strain problem of slope stability in geotechnical engineering. The geometrical model of the slope is shown as Fig.

During the numerical computation, the cubic spline function is chosen as kernel function Eq.(

. CONCLUSION

In addition, it should be mentioned that parallel computation was used for complete comparison between FEM and RKPM on an SGI Onyx3900 supercomputer with 1, 2, 4, 8 and 16 processors utilized.

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Open peer comments: Debate/Discuss/Question/Opinion

<1>Daniel@University of Florida<kjha001@fiu.edu>2013-07-21 13:49:32

Very good