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International Journal of Pressure Vessels and Piping 86 (2009) 711–718

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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Analysis of elastic–plastic problems using edge-based smoothed finiteelement method

X.Y. Cui a,b, G.R. Liu b,c, G.Y. Li a,*, G.Y. Zhang c, G.Y. Sun a

a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, PR Chinab Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1,117576 Singapore, Singaporec Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, 117576 Singapore, Singapore

a r t i c l e i n f o

Article history:Received 5 October 2008Received in revised form8 December 2008Accepted 10 December 2008

Keywords:Numerical methodsMeshfree methodsElastic–plastic analysisFEMES-FEM

* Corresponding author. Tel.: þ86 731 8821717; faxE-mail address: gyli@hnu.cn (G.Y. Li).

0308-0161/$ – see front matter � 2008 Published bydoi:10.1016/j.ijpvp.2008.12.004

a b s t r a c t

In this paper, an edge-based smoothed finite element method (ES-FEM) is formulated for stress fielddetermination of elastic–plastic problems using triangular meshes, in which smoothing domains asso-ciated with the edges of the triangles are used for smoothing operations to improve the accuracy and theconvergence rate of the method. The smoothed Galerkin weak form is adopted to obtain the discretizedsystem equations, and the numerical integration becomes a simple summation over the edge-basedsmoothing domains. The pseudo-elastic method is employed for the determination of stress field andHencky’s total deformation theory is used to define effective elastic material parameters, which aretreated as field variables and considered as functions of the final state of stress fields. The effective elasticmaterial parameters are then obtained in an iterative manner based on the strain controlled projectionmethod from the uniaxial material curve. Some numerical examples are investigated and excellentresults have been obtained demonstrating the effectivity of the present method.

� 2008 Published by Elsevier Ltd.

1. Introduction

For the analysis and design in engineering structures, the elasto-plastic behavior of structure materials needs often to be considered.However, the complicated nonlinear stress–strain relationship andthe loading path dependency in the plastic range make the analysistedious. In the past several decades, the finite element method hasbeen well developed and used as an important tool to analyzematerial nonlinear problems in practical engineering applications[1–4]. However, the displacement-based fully compatible finiteelement method has an inherent characteristic known as theoverly-stiff phenomenon, especially when linear triangularelements are used.

To overcome the overly-stiff phenomenon and effectively‘‘soften’’ the discretized system, Liu et al. have applied thesmoothing technique [5] in a number of meshing free and finiteelement settings. A generalized gradient smoothing technique [6]has been proposed and used to establish weakened weak (W2)formulations known as the generalized smoothed Galerkin weakform [7] that allows the use of discontinuous shape functions. Some

: þ86 731 8822051.

Elsevier Ltd.

important properties including variational consistence, conver-gence, upper bound and soft effects of W2 models have beenrevealed, proved or examined in detail. Liu et al. have also sug-gested various ways (cell-based, node-based, and edge-based) tocreate the smoothing domains for models of desired properties. Inmodels using finite elements, cell-based smoothing domains arecreated by further dividing the elements into one or moresmoothing cells (SC), leading to the so-called smoothed finiteelement method (SFEM) [8–10]. As SFEM computes the integralsalong the edge of the smoothing domains, no derivatives of shapefunctions are needed, no mapping is required, and simple pointinterpolation methods can be used. It works well for very heavilydistorted mesh, in addition to a number of important properties,such as the softening effects, better accuracy and upper bound forsome class of problems. Using the node-based smoothing opera-tion, NS-FEM was also been formulated that can often provideupper bound solutions for force driven problems [11]. Using thepoint interpolation method for shape function construction,a node-based smoothed point interpolation method (NS-PIM or LC-PIM) was formulated [12,13] and extended for heat transfer andthermoelasticity problems [14]. Liu and Zhang [13] proved that theNS-PIM is variationally consistent, can provide much better stressresults, and more importantly can often provide upper boundsolution in energy norm (for force driven problems). It is found,

Edge k

Nodes of the element Centroid of the element

Ωk

Ωk1Ωk2

n1 n2

n3n4

Ωm

n5

n6n7

Fig. 1. The problem domain is divided into Nelement triangular elements with a total ofNedge edges. Interior edge k is sandwiched in the smoothing domain Uk. Smoothingdomain Um for the boundary edge m is a triangle. There are Nk nodes that influence thekth smoothing domain Uk. For domains associated with boundary edges Nk¼ 3; forexample, nodes n5, n6 and n7 influence Um. For domains associated with interior edgesNk¼ 4; for example, nodes n1, n2, n3 and n4 influence Uk.

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718712

however, the NS-FEM or NS-PIM is too soft and hence has spuriousmodes when used for dynamic problems. Recently, an edge-basedsmoothed finite element method (ES-FEM) [15] has been proposedfor 2D solid mechanics problems using edge-based smoothingdomains. It has been found that the ES-FEM model is of closed-to-exact stiffness and gives ultra-accurate (one order more accurate)solution when triangular elements are used compared with the FEM.

A number of numerical techniques has been developed so far tosolve the elasto-plasticity problems. The idea of using elasticsolutions for approximation of inelastic behavior has beenreceiving much interest. Neuber [16] obtained elasto-plastic stressand strains at the stress concentration point using elastic solutionsin early 1960s. Dhalla and Jones [17] used finite element elasticanalysis for approximation of limit loads for pressure vessels andpiping design. Based on this method, Seshadri [18] developeda generalized local stress and strain (GLOSS) method and used it toapproximate plastic strains at local regions. Jahed et al. [19]developed a comprehensive method for solving pressure vesselproblems in the elasto-plastic range based on elastic solutions.Babu and Iyer [20] developed a robust method using relaxationmethod, based on the GLOSS method, and an attempt was made tosatisfy force equilibrium in the plastic range. Chen and Ponterperformed shakedown and limit analyses for 3-D structures [21]and integrity assessment for a tubeplate [22] using linear matchingmethod. This method also applied to the high temperature lifeintegrity of structures [23,24]. Recently, Desikan and Sethuraman[25] proposed a pseudo-elastic finite element method for thedetermination of inelastic material parameters. In this method,material nonlinear problem was solved using the pseudo-elasticlinear finite element method with suitable updation of elasticmaterial properties during the process of iteration. Some researchershave adopted this method to solve material nonlinear problems,such as Sethuraman and Reddy [26], Dai et al. [27] and Gu et al. [28].

In this paper, the edge-based smoothed finite element method(ES-FEM) is formulated for solving material nonlinear problemsbased on Hencky’s deformation theory. The problem domain is firstdiscretized into a set of triangular elements and the smoothingdomains associated with the edges of the triangles are then furtherformed. The material parameters are considered as field variables,and the linear elastic ES-FEM analysis will be carried out to get thepseudo-stress distributions. The stresses in each edge smoothingdomain are constants, and the stresses at the nodes will beobtained by averaging the values of the associated smoothingdomains. An iteration procedure is used to update the materialparameters until equivalent stress–strain point in all smoothingdomains coincide with the uniaxial experimental material curve.The strain controlled projection method is employed to calculatethese effective material parameters. Problems with three materialmodels, elastic-perfectly plastic material, work-hardening materialand Ramberg–Osgood model, are presented to illustrate the effec-tivity of the ES-FEM formulation for the elasto-plastic analysisthrough comparing the numerical results with those obtained bythe finite element commercial software ABAQUS.

2. ES-FEM formulations

As shown in Fig. 1, the problem domain U is divided into Nelement

triangular elements with a total of Nedge edges. Based on thetriangular elements, smoothing domain for each edge is formed bysequentially connecting two end points of the edge and centroids ofits surrounding triangles, such that U ¼ U1WU2W/WUNedge

andUiXUj ¼ B (i s j, i¼ 1,.,Nedge, j¼ 1,.,Nedge). In the ES-FEM, thedisplacement interpolation is element based as in the FEM, but theintegration is based on the smoothing domains that are used forstrain filed smoothing.

At any point in a triangular element, the displacement field u inthe element is interpolated using the nodal displacements at thenodes of the element by the linear shape functions, same as in thestandard linear FEM,

uðxÞ ¼X3

i¼1

NiðxÞdi (1)

where di ¼ fui; vigT is the nodal displacement at node i, Ni(x) isa diagonal matrix of shape functions.

Using strain–displacement equations, the compatible strain ineach element can be given by

3ðxÞ ¼ LuðxÞ (2)

in which

L ¼

v

vx0

v

vyv

vyv

vx0

2664

3775

T

(3)

Substituting Eq. (1) into Eq. (2), we can get

3ðxÞ ¼ BðxÞd (4)

where

B ¼ ½B1;B2;B3�

Bi ¼�

Ni;x 0 Ni;yNi;y Ni;x 0

�T (5)

In order to compensate the ‘‘over-stiffness’’ of the FEM model,a smoothed strain is introduced instead of compatible strain to‘‘soften’’ the system. As shown in Fig. 1, the smoothed strain in thekth smoothing domain can be expressed as [5,6]

3k ¼ZUk

3ðxÞfkðxÞdU (6)

where fk(x) is a given smoothing function that satisfies at leastunity property

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718 713

fkðxÞdU ¼ 1 (7)

ZU

A constant smoothing function is adopted as follows

fkðxÞ ¼�

1=Ak x˛Uk0 x;Uk

(8)

where Ak is the area of the smoothing domain Uk.For interior edges, the smoothing domain Uk of edge k is formed

by assembling two sub-domains Uk1 and Uk2 of two neighboringelements. The sub-domain Uk1 and the sub-domain Uk2 are fromelement e1 and element e2, respectively. The smoothed strain insmoothing domain Uk can be given by

3k ¼1Ak

0B@ Z

Uk1

3k1ðxÞdUþZ

Uk2

3k2ðxÞdU

1CA (9)

where 3k1(x) is the compatible strain calculated in element e1, and3k2(x) is the compatible strain calculated in element e2.

Since linear shape functions are used in the present method, thecompatible strain is a constant in each smoothing sub-domain.Therefore, Eq. (9) can be rewritten as

3k ¼1AkðAk13k1 þ Ak13k2Þ (10)

where Ak1 and Ak2 are areas of the smoothing sub-domains Uk1 andUk2, respectively.

Substituting Eq. (4) into Eq. (10), the smoothed strain can begiven by

3k ¼Ak1

AkBk1dk1 þ

Ak2

AkBk2dk2 ¼ Bkdk (11)

where dk1 and dk2 are the nodal displacements vector of theelement e1 and the element e2, respectively, dk is the displacementvector of the nodes associated with edge k, Bk1 and Bk2 are thecompatible strain matrices of the smoothing sub-domain Uk1 andUk2, respectively.

From Eq. (11), the smoothed strain matrix Bk is written as

Bk ¼Ak1

AkBk1 þ

Ak2

AkBk2 (12)

Note that the sign ‘þ’ denotes assembly but not sum here.Using smoothed strain, the stress in the smoothing domain can

be calculated by

sk ¼ Dkeff 3k (13)

In Eq. (13), Dkeff is the effective material matrix for smoothing domain

Uk, and is obtained from the effective constitutive equation, i.e.

Dkeff ¼

Ekeff

1� nkeff

�

2664

1 nkeff 0

nkeff 1 0

0 0�

1��

nkeff

�2��2

3775 for plane stress

(14)

where Ekeff and nk

eff are effective Young’s modulus and Poisson’sratio, which will be introduced in next section.

Using the smoothed strain obtained previously, we now seek fora weak form solution of displacement field u that satisfies thefollowing smoothed Galerkin weak form [6]

d3Ts dU� duTb dU� duTt dG ¼ 0 (15)

ZU

ZU

ZG

where b is the body force, and t is the boundary traction.Substituting Eqs. (1), (11) and (13) into Eq. (15), a set of dis-

cretized algebraic system equations can be obtained in thefollowing matrix form

Kd� f ¼ 0 (16)

where d is the vector of nodal displacement at all the nodes, and f isthe force vector defined as

f ¼ZU

NTðxÞb dUþZG

NTðxÞt dG (17)

In Eq. (16), K is the (global) smoothed stiffness matrix of presentES-FEM, it is assembled in the form of

Kij ¼XNedge

k¼1

KijðkÞ (18)

The summation in Eq. (18) means an assembly process same as thepractice in the FEM, Nedge is the number of the edges of the wholeproblem domain U, and KijðkÞ is the stiffness matrix associated withUk that is computed by

KijðkÞ ¼ZUk

ðBkÞTi Dkeff ðBkÞj dU ¼ ðBkÞTi Dk

eff ðBkÞjAk (19)

3. Stress–strain relationship for effective material parameters

From the work of Jahed et al. [19], the strain–stress relationshipcan be taken in the form of

3ij ¼ f�sij

(20)

in which total strain 3ij is the summation of an elastic part 3eij and

a plastic part 3pij,

3ij ¼ 3eij þ 3p

ij (21)

The elastic strain tensor relates to the stress tensor by Hooke’slaw for isotropic material

3eij ¼

1þ n

Esij �

n

Eskkdij (22)

where n is Poisson’s ratio, E is Young’s modulus, and dij is the Deltafunction.

The plastic strain tensor is related to the deviatoric componentof stress tensor and is given by Hencky’s deformation theory

3pij ¼ FSij (23)

where

Sij ¼ sij �13

skkdij (24)

and F is a scalar valued function as given by

F ¼33p

eq

2seq¼ 3

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23p

ij3pij=3

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3SijSij=2

q (25)

p

R1R2

a

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718714

Substituting Eqs. (22)–(25) into Eq. (21) yields

3ij ¼�

1þ n

Eþ F

�sij �

�n

Eþ 1

3F

�skkdij (26)

All the variables inside the parentheses in Eq. (26) are involved withthe material properties, final equivalent plastic strain and equiva-lent stress. This equation can be rewritten as

3ij ¼

1þ neff

Eeff

!sij �

neff

Eeff

!skkdij (27)

where Eeff and neff are the equivalent Young’s modulus and Pois-son’s ratio, which are given by

Eeff ¼1

ð1=EÞ þ ð2F=3Þ (28)

neff ¼ Eeff

�n

Eþ F

3

�¼ Eeff

n

Eþ 1

2

1

Eeff� 1

E

!!(29)

4. Determination of effective material parameters

In the present section, projection method [25] is used for thedetermination of effective material parameters, Eeff and neff, neededto calculate Deff. First, a linear elastic analysis is carried out to getthe initial stress field. The equivalent stress using von Mises yield isused in comparison with the yield stress s0. If the equivalent stressis smaller than the yield stress s0, the computing is completedbecause the material is still in the elastic region; if the equivalentstress is larger than the yield stress s0, it means that the defor-mation has already entered the plastic region, and the followingiteration will be performed.

From the ES-FEM linear elastic analysis, we get the equivalentstress for each smoothing domain, and the state is shown as point 1in Fig. 2. Keeping the strain value 31 the same (i.e. strain controlled),and projecting point 1 on the experimental uniaxial material curveto get point 10, the effective value of Young’s modulus, Eð1Þeff , for thenext iteration is obtained from the slope of the straight line 0–10.Substituting this effective value into Eq. (29), the effective Poisson’s

εε

E

Eeff(1)

1

1'

2

2'

3

3'

0

Eeff(2)

(2)ε (1)

Fig. 2. Projection method for determination of Eeff.

ratio, nð1Þeff , can also be obtained. With the new effective materialparameters the next ES-FEM linear elastic analysis is performed toget point 2 and its projection 20, and further to obtain Eð2Þeff and nð2Þeff .This iterative procedure is repeated until all the effective materialparameters converge and equivalent stresses of all points fall on theexperimental uniaxial stress–strain curve. The convergence ischecked using following criterionffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNedge

k¼1

�Ek

eff ðiþ 1Þ � Ekeff ðiÞ

�2

PNedge

k¼1

�Ek

eff ðiÞ�2

vuuuuut � d (30)

where Ekeff ðiÞ and Ek

eff ðiþ 1Þ are the effective Young’s modulus of theith and (iþ 1)th iteration steps of the kth smoothing domain,

b

Fig. 3. Cylindrical pressure vessel subjected to internal pressure; (a) geometry and theboundary loading conditions; (b) mesh arrange of the model.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-1

-0.5

0

0.5

1

R

Nor

mal

ized

str

esse

s

Lines:ABAQUS (Quad)Nodes:ES-FEM (Tri)σEq/σ0

σθ/σ0

σr/σ0

P/σ0 = 1.0

Fig. 4. Normalized stress distributions for elastic-perfectly plastic material. Lines:ABAQUS, nodes: ES-FEM.

0 1 2 3 4 5

x 10-3

0

0.5

1

1.5

2

2.5x 108

ε (m/m)

σ (P

a)

Uniaxial material curve

State of stress for a set of material points

Fig. 5. State of stress for a set of nodes in radial direction after convergence for elastic-perfectly plastic material.

0.1 0.2 0.3 0.4 0.5

-1

-0.5

0

0.5

1

P/σ0 = 1.0

P/σ0 = 0.8

P/σ0 = 0.575

σEq/σ0

σθ/σ0

σr/σ0

P/σ0 = 1.125

Lines:ABAQUS(Quad)

Nodes:ES-FEM(Tri)

R

Nor

mal

ized

str

esse

s

Fig. 6. Normalized stress distribution for different pressure ratios for elastic-perfectlyplastic material. Lines: ABAQUS, nodes: ES-FEM.

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718 715

respectively, and d is the tolerance for convergence constant whichis set to 10�3 in the study of the numerical examples.

It must be pointed out that if the applied loading is just largeenough for the results failing to converge, the material is thenregarded failed, and this loading is marked as the critical failureloading.

In this study, three different material models, elastic-perfectlyplastic material, linearly work-hardening material and Ramberg–Osgood model, will be investigated for numerical examples. Forelastic-perfectly plastic material, the stress–strain relation isgiven by

3 ¼�

s=E s < s0s0=E þ 3p s � s0

(31)

Using Eq. (28), Eeff can be expressed by

Eeff ¼s0

3(32)

In the case of linearly work-hardening material model, it isassumed that the material has tangent modulus ET. The materialcurve is

3 ¼�

s=E s < s0s0=E þ ðs� s0Þ=ET s � s0

(33)

and Eeff is given by

Eeff ¼s0 þ 3pET

3(34)

Ramberg–Osgood model is one general case of hardeningmaterial, and is described by the following formulation

3

30¼ s

s0þ a

�s

s0

�n

(35)

where 30¼ s0/E is the strain at initial yield, a is the yield offset, andn is the hardening exponent. Both a and n are preassigned materialconstants before computation. The effective Young’s modulus Eeff isobtained as

Eeff ¼ 1.�1

Eþ a

30

s0

�s

s0

�n�1�(36)

For a determined strain, the stress state according to the straincan be calculated from Eq. (35) using a nonlinear equation solver.Eeff can then be evaluated from Eq. (36). Once Eeff is determined, neff

can be obtained from Eq. (29).

5. Numerical examples

5.1. Cylindrical vessel

To illustrate the validity of the proposed ES-FEM in materialnonlinear problems, a cylindrical vessel under plane stress condi-tions subjected to an internal pressure P is investigated. Thegeometry and the boundary loading conditions are shown inFig. 3a. The inner radius is R1¼0.1 m, and the outer radius isR2¼ 0.5 m. The material properties are taken as Young’s modulusE¼ 2.0�1011 Pa, Poisson ratio n¼ 0.3, and yield stresss0¼ 2.0�108 Pa. For linearly work hardening case, the tangentmodulus is taken as ET¼ E/4. For Ramberg–Osgood model, the yieldoffset a¼ 3/7 and hardening exponent n¼ 5 are considered. Owingto the symmetry conditions, only a quarter of the cylindrical vesselis modeled, and the model is divided into 20�10 elements, asshown in Fig. 3b.

To validate the accuracy of the present solutions, the analysisusing the finite element commercial software ABAQUS is alsocarried out using quadrilateral element with the same discretiza-tion nodes. At first, the material is considered as elastic-perfectlyplastic model. The variations of radial, hoop and equivalent vonMises stresses along the thickness direction of the cylinder forpressure ratio P/s0¼1.0 are shown in Fig. 4. In present ES-FEM, thestress values at node are obtained by averaging the values of the

0 0.5 1 1.5 2 2.5 3 3.5

x 10-3

0

0.5

1

1.5

2

2.5

3

3.5x 108

ε (m/m)

σ (P

a)

Uniaxial material curveConvergence path

Fig. 7. The convergence path for a particular point for elastic-perfectly plastic material.

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

1

2

3

4x 108

ε (m/m)

Uniaxial material curveConvergence path

σ (P

a)

Fig. 10. The convergence path for a particular point for linearly hardening materialmodel.

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718716

associated smoothing domains. It is observed that the presentstresses’ nodes consist well with ABAQUS curve all along. Fig. 5demonstrates the state of von Mises stress for a set of nodes inradial direction after convergence. It can be seen that all the nodesare in good agreement with the uniaxial material curve. Fig. 6 givesthe stress variations for different internal pressure ratios. Thedimension of the plastic zone can be easily estimated from stressdistributions, which compares well with ABAQUS quadrilateral

0 1 2 3 4

x 10-4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Radial displacement at inner surface (mm)

Pre

ssur

e / Y

ield

str

ess

Reference (ABAQUS 40000 Quad elements)ES-FEM (400 Tri elements)

Fig. 8. The radial displacement at inner surface with different pressure using elastic-perfectly plastic material.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

-1

-0.5

0

0.5

1

1.5

R

Nor

mal

ized

str

esse

s

P/σ0 = 1.0

σEq/σ0

σθ /σ0

σr /σ0

Lines:ABAQUS (Quad)Nodes:ES-FEM (Tri)

Fig. 9. Normalized stress distributions for linearly hardening material model. Lines:ABAQUS, nodes: ES-FEM.

element. As elastic-perfectly plastic model is used, the von Misesstresses in plastic zone are all equal to yield stress s0.

Fig. 7 shows the convergence path for a material point during theprocess of iteration. Here, convergence is assumed to be achievedwhen Eq. (30) is satisfied. It can be seen that the point falls on theuniaxial material curve quickly. Fig. 8 presents the load–displacement

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1

-0.5

0

0.5

1

1.5

R

Nor

mal

ized

str

ess

σEq/σ0

σθ/σ0

σr/σ0

P/σ0 = 1.0

Lines:ABAQUS (Quad)Nodes:ES-FEM (Tri)

Fig. 11. Normalized stress distributions for Ramberg–Osgood material model. Lines:ABAQUS, nodes: ES-FEM.

0 1 2 3 4 5 6 7x 10-3

0

1

2

3

4x 108

ε (m/m)

σ (P

a)

Uniaxial material curveConvergence path

Fig. 12. The convergence path for a particular point for Ramberg–Osgood materialmodel.

15

R=5mm

60 p

Or

z

0

p

p

50

45

40

30

25

15

10

0

10 20 25 30 355

5

20

35

55

35mm

60mm

Fig. 13. A nozzle subjected to internal pressure, geometry and the boundary loading conditions.

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718 717

curves obtained by the present method. For comparison, the referencesolutions obtained using ABAQUS with large number of quadrilateralelements (40,000) are also plotted in the same figure. It is obvious thatthe ES-FEM method gives very accurate results.

The example is performed again using the linearly hardeningmaterial model. The distributions of the von Mises stress, hoopstress and radial stress are all shown in Fig. 9. It clearly shows thatthe solutions using present ES-FEM coincide well with those ofABAQUS quadrilateral element. We can easily find that plasticdeformation occurs in the region in which the normalized equiv-alent von Mises stress is larger than 1.0. Because of the materialwork hardening, the normalized equivalent von Mises stress inthe plastic region no longer remains at 1.0. Fig. 10 shows theconvergence path for a particular point using linearly hardening

3403203002802602402202001801601401201008060

ES-FEM, 1058 nodes ABAQUS, Q4,368

Fig. 14. Comparison of computed v

material model. It can be observed that all the nodes fall on theuniaxial material curve quickly.

A Ramberg–Osgood material model with yield offset a¼ 3/7 andhardening exponent n¼ 5 is also considered for this problem. Theresults are shown in Figs. 11 and 12. It is observed again that thepresent solutions match well with those of ABAQUS quadrilateralelement and the final stress–strain state as well as the convergencepath are quite reasonable.

5.2. A nozzle with internal pressure

A nozzle subjected to internal pressure is analyzed to demon-strate more features of the present method. The geometry and thenumerical model are shown in Fig. 13. The same case is also

3403203002802602402202001801601401201008060

ABAQUS, T3,1058 nodes68 nodes

3403203002802602402202001801601401201008060

on Mises stress distributions.

X.Y. Cui et al. / International Journal of Pressure Vessels and Piping 86 (2009) 711–718718

analyzed using ABAQUS with triangular element and a referencesolution is also computed using ABAQUS quadrilateral elementswith large number of nodes (36,868) for comparison. Linear work-hardening material model is employed here, and the materialparameters are given as Young’s modulus E¼ 2.1�105 MPa, Pois-son ratio n¼ 0.3, yield stress s0¼ 210 MPa, and the tangentmodulus is taken as ET¼ E/4. The internal pressure p is equal to theyield stress s0.

Fig. 14 shows the comparison of computed von Mises stressdistributions between the present method, ABAQUS using the sametriangular elements and the reference ones obtained using ABAQUSwith fine mesh (36,868) of quadrilateral elements. It is clearlyshown that the present result agrees better with reference solutionthan those obtained using ABAQUS triangular elements with thesame mesh, especially in the zone of large von Mises.

6. Conclusions

In this paper, the edge-based smoothed finite element method(ES-FEM) is formulated to analyze material nonlinear problems. Inpresent ES-FEM, the smoothed Galerkin weak form is used for dis-cretizing the system equations and the numerical integration isperformed based on the smoothing domains associated with edges ofthe mesh. Material nonlinearity is considered as pseudo-linear elasticanalysis by suitable updating of material properties in terms ofeffective material parameters. Based on Hencky’s total deformationtheory, the effective elastic material parameters can be easilyobtained in an iterative procedure from the one-dimensional uniaxialmaterial curve. Numerical examples using von Mises material havebeen successfully analyzed obeying elastic-perfectly plastic, linearlywork-hardening or Ramberg–Osgood hardening model, respectively,and very good results have been obtained. Through these investiga-tions, the following conclusions can be drawn.

(1) In the present ES-FEM, the formulation is straightforward andthe implementation is as easy as the FEM, without the increaseof degree of freedoms. Hence the present method is verysimple and can be easily implemented with little changes tothe FEM code.

(2) Through smoothing operation, the present method can providea much needed softening effect to the model and the ‘‘overly-stiff’’ phenomenon of the compatible displacement-based FEMmodel is ameliorated effectively. Therefore, the performance ofthe present method is greatly enhanced, and numerical resultsobtained using triangular elements achieve the same accuracylevel as ABAQUS quadrilateral elements.

(3) In the proposed method, many techniques used in linear elasticanalysis can be easily incorporated here with only minorrevisions. Compared with the conventional inelastic analysisusing classical incremental theory and Newton–Raphsonmethod, the present scheme can be easily implemented ina numerically straightforward way.

Acknowledgements

The support of National Outstanding Youth Foundation(50625519), Key Project of National Science Foundation of China(60635020), Program for Changjiang Scholar and InnovativeResearch Team in University and the China-funded Postgraduates’Studying Abroad Program for Building Top University are gratefullyacknowledged. The authors also give sincerely thanks to thesupport of Centre for ACES, Singapore-MIT Alliance (SMA), andNational University of Singapore.

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