Simulation of Plane Strain Rolling Through a Combined Element Free Galerkin-Boundary Element...

Journal of Materials Processing Technology 159 (2005) 214–223

Simulation of plane strain rolling through a combined elementfree Galerkin–boundary element approach

Xiong Shangwu∗,1, J.M.C. Rodrigues, P.A.F. MartinsDepartamento de Engenharia Mecˆanica, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Received 24 June 2002; accepted 6 May 2004

Abstract

This paper presents an innovative approach for analysing plane strain rolling. The proposed approach is based on a solution resulting fromthe combination of the element free Galerkin method (EFG) and the boundary element method (BEM). The element free Galerkin method isused to perform the numerical modelling of the workpiece allowing the estimation of the roll separating force, the rolling torque and the con-tact pressure along the surface of the rolls, whilst the boundary element method is applied for computing the elastic deformation of the rolls.

Coupling between the two numerical methods is made using the element free Galerkin estimate of the contact pressure along the surfaceof the rolls to define the boundary conditions to be applied in the elastic analysis of the rolls. The validity of the proposed approach isdiscussed by comparing the theoretical predictions with experimental data found in the literature.© 2004 Elsevier B.V. All rights reserved.

Keywords:Plane strain rolling; Element free Galerkin method; Boundary element method

1. Introduction

Taking a general view of the present state of the art it ap-pears that the finite element method is the most well estab-lished approach for the numerical simulation of metal form-ing processes. The appeal of the finite element method stemsfrom its ability of being easily implemented in the form ofcomputer programs that can systematically represent mate-rial behaviour and complex boundary conditions.

However, the method can sometimes experience severelimitations in handling non-stationary large plastic defor-mations due to the progressive distortion of the mesh andits eventual interference with the dies. Therefore, in ad-dition to the finite element method several automatic andadaptive remeshing techniques have also been develop tocircumvent the aforementioned difficulties. Unfortunately,the remeshing procedures may contribute to a degradationof the overall accuracy and to an increase of computa-tional efforts namely during three-dimensional analysis ofcomplex forming processes.

∗ Corresponding author. Fax:+351 21 8419058.E-mail address:[email protected] (X. Shangwu).

1 Present address: Department of Mechanical Engineering, NorthwesternUniversity, 2145 Sheridan Rd, Evanston, IL 60208, USA.Fax: +1-847-4673915.

Recent developments in meshless (or meshfree) methods[1–8] and their expected evolution[9–12] seem promisingin overcoming some of the above-mentioned problems. Infact, meshless methods perform the discretization of theworkpiece entirely in terms of arbitrarily placed nodeswithout the use of an explicit mesh. Another potential ad-vantage of meshless methods is the possibility of handlingmore easily the procedures of refinement and adaptivityfrequently utilised in the numerical simulation of metalforming processes.

The first attempts to solve metal forming problems usingmeshless methods were made by Chen et al.[13–15] whoused the reproducing kernel particle method (RKPM)[5]to simulate plane strain stretching of a circular blank andsimple axisymmetrical upsetting operations. Elastic-plasticmaterial models were employed and a general good agree-ment was found between theoretical results and experimen-tal data. Kulasegaram and co-workers[16,17] employedthe corrected smooth particle hydrodynamics method(CSPH) to perform several two-dimensional simulationsof basic metal forming processes. The method was ap-plied to plane-strain extrusion, rolling and upsetting, butonly perfectly-plastic materials (i.e. without strain hard-ening effects) were taken into account. The validation oftheoretical predictions against experimental data or al-ternative well-established numerical methods was limitedto simple cases. More recently, Shangwu et al.[18,26]

0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2004.05.008

X. Shangwu et al. / Journal of Materials Processing Technology 159 (2005) 214–223 215

utilised the element free Galerkin method (EFG)[4] andRKPM to simulate flat rolling using a slightly compressiblerigid-plastic material model. The methods were utilisedfor predicting the roll separation force and torque aswell as the distribution of the major field variables un-der the hypothesis of plane strain deformation with rigid(non-deformable) rolls. It was concluded that both EFGand RKPM were able to provide theoretical predictionsin good agreement with experimental data and that majordiscrepancies were probably due to the assumption of rigidrolls.

The present paper draws from the previous investiga-tions on the applicability of EFG to the numerical sim-ulation of flat rolling to the development of an innova-tive computational approach founded on the combinationof the element free Galerkin method with the boundary el-ement method (BEM). The element free Galerkin methodis used to perform the numerical modelling of the rolledmaterial whilst the boundary element method is appliedfor the elastic analysis of the rolls. Coupling between thetwo methods is made using the EFG solution of the con-tact pressure along the surface of the rolls to define theboundary conditions to be applied in the analysis of therolls.

The main advantage of the proposed approach over thealternative meshless solution, requiring the complete dis-cretization of both the workpiece and the rolls, is the neces-sity of defining only the contour of the rolls, simplifying theanalysis and offering additional computational advantages.Furthermore, the calculation of the main field variables in-side the rolls (e.g. strains and stresses) can be left out ofthe numerical simulation procedure being only performed atthe post-processing stage. The usefulness and accuracy ofthe proposed EFG–BEM combined approach is discussedby comparing theoretical predictions with experimental datafound in the literature[19].

2. Theoretical background

The element free Galerkin method developed by Be-lytschko et al.[4] is based on the diffuse element method(DEM) previously proposed by Nayroles et al.[3]. So far,the method has been mainly utilised for solving linear elas-tic problems related to fracture and crack growth[4,11,12].The aim of this section is to describe how the method canbe extended to metal forming by presenting an approachbased on the slightly compressible rigid-plastic formulation.

The theoretical model, to be derived in this section, wasdeveloped under the assumption of steady-state plane strainrolling of rigid plastic materials with elastically deformablerolls. The rolls have identical radius,R, surface conditions,applied power and circumferencial velocityU. The rollingprocess is schematically shown inFig. 1. Due to symmetryof the deformation, only the upper half of the workpiece isutilised in the analysis.

2.1. Element free Galerkin interpolant

Numerical procedures involving differential equationsusually perform the approximationuh(x) for the unknownfunction u(x) in a domain ∈ Rd , (d is the number of di-mensions of the problem), by mean of trial functions whichcan be expressed in terms of base functions with unknowncoefficients[4],

uh(x) =m∑i=1

pi(x)ai(x) = pT(x)a(x) (1)

In the above equationm represents the number of indepen-dent base functions (e.g. monomials or Legendre polyno-mials),p(x) is the vector of base functions anda(x) is thevector of unknown coefficients which can be obtained byminimising the following norm,

J(a(x)) =n∑

I=1

w(x − xI)[pT(xI)a(x) − uI ]

2 (2)

whereuI is the nodal value for the functionu(x) at x = xIandn is the number of points in the neighbourhood ofx forwhich the kernel or weight functionw(x − xI) > 0. Thisimplies that the unknown coefficientsai can be determinedfrom,

∂J(a(x))

∂a= 2

n∑I=1

w(x − xI)p(xI)[pT(xI)a(x) − uI ] = 0

(3)

giving rise to the following relationship,

a(x) = A−1(x)B(x)u (4)

In the above equation the matricesA(x) and B(x) can beobtained from,

A(x) =n∑

I=1

wI(x)p(xI)pT(xI), wI(x) ≡ w(x − xI);

B(x) = [ w1(x)p(x1) w2(x)p(x2), . . . , wn(x)p(xn) ];uT = [ u1 u2, . . . , un ] (5)

Rewriting the approximationuh(x) for the unknown func-tion u(x) in (1) as,

uh(x) =n∑

I=1

φI(x)uI (6)

and comparing the aforementioned relationship with theclassical finite element interpolants it can be concluded thatthe meshless shape functions of the moving least squaresapproximation methods can generally be written as[4],

φI(x) =m∑j=1

pj(x)(A−1(x)B(x))jI (7)

At this stage it is of interest to note that in meshless meth-ods, unlike finite elements, the shape functions are not

216 X. Shangwu et al. / Journal of Materials Processing Technology 159 (2005) 214–223

Fig. 1. (Top) Detail of the cell structure utilised in the element free Galerkin analysis of the rolled workpiece. (Bottom) Schematic illustration ofthecontrol volume utilised in the combined EFG–BEM numerical analysis of flat rolling under plane strain conditions.

interpolants (φI(xJ ) = δIJ) and therefore the value of theapproximating function generally does not coincide withthe value of the unknown parameter at the sampling points,uh(xI) = uI . This creates extra difficulties in enforcing theessential boundary conditions. However, there are severaldeveloped techniques to overcome this difficulty[11,12].

The partial derivatives of the shape functionsφI(x) canbe obtained applying the usual rules of partial differenti-ation (the index following a comma represents a spatialderivative),

φI,i =m∑j=1

pj,i(A−1B)jI + pj(A−1,i B + A−1B,i)jI (8)

where,

A−1,i = −A−1A,iA

−1 (9)

By selecting a vectorp(x) of linear base functions, theexplicit form of the meshless shape functions inEq. (7)becomes similar to the kernel with first degree of correctionthat was proposed by Liu et al.[5,9] in the reproducingkernel particle method,

φI(x) = wI(x)α(x)(1 + β(x)T(x − xI)) (10)


where the termsα(x) andβ(x) can be evaluated by[18],

α(x)= 1∑nI=1wI(x)(1 + β(x)T(x − xI))

,

β(x)=[

n∑I=1

(x − xI) ⊗ (x − xI)wI(x)

]−1

×n∑

I=1

(xI − x)wI(x) (11)

wherex ⊗ y = xiyj (i, j = 1, . . . , d ).Accordingly, it follows that the spatial derivatives of the

shape function can be evaluated as,

∇φI(x)= (∇wI(x)α(x) + wI(x)∇α(x))[1+β(x)T(x − xI)]

+wI(x)α(x)[(∇β(x))T(x − xI) + Iβ(x)] (12)

where,

∇α(x) =−∑n

I=1∇wI(x)(1 + β(x)T(x − xI))

+wI(x)∇(1 + β(x)T(x − xI))(∑nI=1wI(x)(1 + β(x)T(x − xI))

)2,

∇β(x) =[

n∑I=1

(x − xI) ⊗ (x − xI)wI(x)

]−1

×

n∑I=1

∇(xI − x)wI(x)+n∑

I=1

(xI − x)∇wI(x)

−∇[

n∑I=1

(x − xI) ⊗ (x − xI)wI(x)

]β(x)

(13)

2.2. Solution for the workpiece

The analysis of the rolled workpiece follows the slightlycompressible rigid-plastic flow formulation, which was pro-posed by Osakada et al.[20],

σij ,j = 0 (14)

εij (u) = 1

2

(∂ui

∂xj+ ∂uj

∂xi

)= 1

2(ui,j + uj,i) (15)

σij = σ

˙ε

(2

3εij +

(1

g− 2

9

)εkkδij

)(16)

whereσ =√(2/3)σ′

ijσ′ij + g(σkk/3)2 is the effective stress,

εij (u) is strain-rate tensor,˙ε(u) =√(2/3)εij (u)εij (u) + ε2

kk/g

is the effective strain-rate,σ′ij = σij − σkk/3δij is the de-

viatoric stress tensor,δij is Kronecker delta andg is asmall positive constant[20] (generally varying from 0.01 to0.0001).

In present study, the following essential velocity boundaryconditions on the surfaceSu are considered (refer toFig. 1):

Line AB:ux = ux0 = constant(unknown),uy = 0

Line CD:ux = ux1 = constant(unknown)uy = 0

Line AD: uy = 0 Line BC:uini = 0

wheren = [ sinθ cosθ ]T andθ is the angular position withreference to a vertical axis crossing the exit of the plasticdeformation zone.

Furthermore, the traction boundary condition onSf (lineBC) is,

σti = σijnj − ((σklnl)nk)ni = τfi = −|τf | ui

|$uf | (17)

whereσti , is the tangential component of stress atSf ,$uf =√ui · ui is the norm of the relative slipping velocity,ui =

(u−utool)i, along the roll-workpiece contact interface wherethe friction shear stressesτfi are applied,uT

tool = (U cosθ−U sinθ) is the velocity vector of the roll withuT

tooln = 0.Hence, the variational principle associated with the solu-

tion of rigid-plastic material deformation problems consid-ers the velocityu(x) ∈ H1, (H1 is Soblev spaces of degreeone), to be the solution only if it corresponds to the absoluteminimum of the total potential energyE(u),

E(u)=∫Ω

σ(u) ˙ε(u)dΩ +∫Sf

|τf | · |$uf | dS

+ 1

2ku

∫Su

u2y dS + 1

2kf

∫Sf

(uini)2 dS (18)

In the above equationku andkf are penalty functions whichare introduced to enforce the above essential boundary con-ditions. It is worth notice that meshless methods demand theutilisation of penalty functions[11] or other related tech-niques[12] because, unlike finite elements, the trial func-tions do not satisfy the essential boundary conditions.

The simplest procedure to satisfy the equilibriumEq. (14)and to satisfy boundary conditions is to express the weakform of the variational principle (18) entirely in terms of anarbitrary variation in the velocityv ∈ H1 as,∫Ω

σij (u)εij (v)dΩ + ku

∫Su

uyvy dS −∫Sf

σtivi dS

+ kf

∫Sf

uini · vjnj dS = 0, ∀v ∈ H1 (19)

where, (refer toEqs. (6) and (10))

u(x) =n∑

I=1

φI(x)uI , v(x) =n∑

I=1

φI(x)vI (20)

It is of interest to note that in the case of finite elementsφI = NI is the shape function of the element and n is thenumber of nodal points defining the element. In other words,the domain of interpolation of finite elements is limited toeach element utilised in the discretization of the workpiece.


Following the conventional discretization procedure, thedistribution of strain-rate (15) inside the workpiece can beapproximated by,

ε(u) = Bu (21)

whereB is the strain-rate matrix,

BTI =

[φI,x 0 φI,y0 φI,y φI,x

](22)

Thus distribution of stress (16) can be obtained from,

σ(u) = σ

˙ε

(D + 1

gCCT

)Bu (23)

whereC represents the vectorial form of the Kronecker sym-bol, and the matrixD is,

D =

4

9

−2

90

−2

9

4

90

0 01

3

(24)

SubstitutingEqs. (20)–(24)into the weak form of the vari-ational principle (19), it is possible to obtain the followingset of non-linear equations,

Ku = f (25)

where,

KIJ =∫Ω

σ

˙εBTI

(D + 1

gCCT

)BJ dΩ

+ku

∫Su

φI

[0 00 1

]φJ dS

+∫Sf

((|τf | I

|$uf | + kfnTn

)φIφJ dS (26)

and,

f I =∫Sf

φI |τf | utool

|$uf | dS (27)

In the above equations, the sense of the friction shear stressτf is opposed to the relative slipping velocity,$uf , accord-ing to,

τf = −mfk$uf

|$uf | , k = σ√3

(28)

wheremf is the friction factor following the law of constantfriction.

In order to avoid numerical problems inEqs. (26)and (27)due to abrupt changes in the friction shear stress atthe neutral point, the following alternative form ofEq. (29)was implemented[21],

τf = −mfk

2

πarctan

( |$uf |u0

)$uf

|$uf | , k = σ√3

(29)

whereu0 is an arbitrary value much smaller than the normof relative slipping velocity,$uf , is given by,

|$uf | =√(ux − U cosθ)2 + (uy + U sinθ)2 + e1 (30)

where,e1, is a small positive constant to prevent numericaldifficulties at the vicinity of the neutral point[22].

The numerical evaluation of the integrals included in (26)and (27) is performed in accordance to the cell structuredepicted inFig. 1. The nodes are located on the material flowlines and four neighbouring nodal points form a cell. In orderto ensure the incompressibility requirements of the plasticdeformation of metals both complete (2× 2) and reduced(1 × 1) Gauss point integration schemes are utilised for thecalculation of the domain integral. The numerical integrationof the boundary integrals is performed by means of a 5point Gauss quadrature. By introducing the different typesof Gauss point quadrature schemes in the system ofEq. (25),considering a workpiece that is discretized by NP nodes anddenoting byc, a Gauss quadrature point withinΩ and byB,F, Gauss quadrature points onSu, andSf , respectively, it ispossible to obtain an explicit form ofEqs. (25)–(27)readyfor computer implementation,

NP∑J=1

KIJuJ = f I , (I = 1,2, . . . ,NP) (31)

where,

KIJ =∑c∈MIJ

−2µc

3(∇φI(xc) ⊗ ∇φJ(xc))

+µc(∇φI(xc) · ∇φJ(xc))I

+µc(∇φJ(xc) ⊗ ∇φI(xc))

|Jc|

+∑b∈Mb

IJ

3µb

g(∇φI(xb) ⊗ ∇φJ(xb))|Jb|

+∑

F∈MFIJ

(|τf (xF )| I

|$uf (xF )| + kfnTFnF

)

×φI(xF )φJ(xF )|JF | + ku∑B∈MB

IJ

φI(xB)

[0 00 1

]

×φJ(xB)|JB| (32)

and,

f I =∑

F∈MFI

φI(xF )|τf (xF )| utool(xF )

|$uf (xF )| |JF | (33)

whereµ = σ/3˙ε, andλ = (σ/ ˙ε)((1/g) − (2/9)), NC, NF,NB are the total number of Gauss quadrature points respec-tively in Ω, Sf andSu, Mb

IJ is the set of Gauss pointsb lo-cated within the overlapped domain of bothxI andxJ withwI(xb) > 0 andwJ(xb) > 0, which is used for performingthe reduced integration of the domain integral in (32).


Similarly the total potential energy can be obtained from,

E(u) =NC∑c=1

σc(u) ˙εc(u)|Jc| +NF∑F=1

kf

2(uFnF )

2

+ |τf (xF )| · |$uf (xF )|

|JF | + ku

2

NB∑B=1

(uy)2B|JB|

(34)

The setting of unknowns is performed in the same way asin finite elements whilst the essential boundary conditionsare imposed by a penalty approach directly applied over theGauss interpolation points during the numerical integrationprocedure. Therefore, the approximation of the velocity fieldfor each Gauss integration point will be given by,

u(x) ∼=NB∑I=1

φI uI +Nnod∑J =I

φJuJ (35)

whereNB is the number of nodes with essential boundaryconditions,Nnod is the number of interior nodes and NP=NB +Nnod is the total number of nodes utilised in the mesh-less discretization of the workpiece.

The system ofEq. (31)is ready for being solved with thehelp of a computational scheme based in the direct iterationmethod, and the present work makes use of the G-functionmethod suggested by Mori et al.[23] to set up an initial guessof velocity field. However, it is possible to linearize it usinga Newton–Raphson procedure with the aim of achieving afaster convergence. The technique to be employed is similarto that used in finite element analysis, and can be written as,

Ψ(l+1) = Ψ(l) +(∂Ψ

∂u

)(l)

$u(l+1) = 0,

ΨI = ∂E(u)

∂uI

=NP∑J=1

KIJuJ − f I (36)

where the subscriptl denotes the number of the iteration and,

NP∑a=1

∂(ψI)i

∂(ua)k$(ua)k

=NP∑J=1

∑c∈MIJ

[ωc((σc)mkφJ,m(xc)) · (σc)ijφJ,j(xc)µc

× ∇φJ(xc)·∇φI(xc)δik+(λc+µc)φJ,i(xc)φJ,k(xc)]|Jc|+

∑B∈MB

IJ

kuφI(xB)φJ(xB)δik|JB|+∑

F∈MFIJ

[ |τf (xF )||$uf (xF )|3

× (|$uf |2δik − uiuk) − nink

]φI(xF )φJ(xF )|JF |

×$(uJ)k (37)

In the above equationωc = (∂σc/∂ ˙εc − σc/ ˙εc)/σ2c ,

i, j, k,m = 1, . . . , d for a d-dimensional problem and therepeated indices of the footnotes indicate summation.

To solve the system of equations resulting from (31) itis necessary to establish an iterative procedure in which thenodal velocityu between two successive iterations is relatedwith the perturbation value$u by

u(l+1) = u(l) + ϕ$u(l+1) (38)

whereϕ is the deceleration coefficient. Several criteria canbe utilised for checking the overall quality of the approxima-tion: (i) the error norm of the velocity||$u||/||u||; (ii) thenorm of the residual ||Ψ || and (iii) the ratio of total potentialenergyE(u) between two successive iterations. Generally itis required that each of the criteria decreases from iterationto iteration.

After solving the system ofEq. (31), the separating forceP and the rolling torqueQ can be approximately obtained as,

P =∫Sf

σy dS (39)

Q = Φ

υ= ΦR

U(40)

whereυ is angular velocity of the roll andΦ is the totalpower required by the rolling process (i.e. the sum of plasticdeformation rate and friction energy rate),

Φ =∫Ω

σ ˙εdΩ +∫Sf

|τf | · |$uf |dS (41)

2.3. Solution for the rolls

The analysis of the rolls is treated separately using theboundary element formulation for linear elastostatics; bodyforces being neglected. Under these circumstances theboundary integral equation to a particular point ‘i’ is givenby

cilkwik =

∫S

tkω∗lk dS −

∫S

ωkt∗lk dS (42)

whereSdenotes the contour of the roll,ω∗lk andt∗lk are known

functions (the so-called fundamental solutions) for the dis-placements and tractions in thek direction due to a unit loadin the l direction acting at the source point ‘i’. The arraycilkis a multiplier (view factor) that depends only on the geome-try of the problem at ‘i’; smooth surfaces givingcilk = δlk/2.The fundamental solution for two dimensional plane strainis expressed as[24]:

ω∗lk = 1

8πG(1 − ν)

[(3 − 4ν)ln

(1

r

)δlk + ∂r

∂xl

∂r

∂xk

],

t∗lk = 1

4π(1 − ν)r

[∂r

∂n

(1 − 2ν)δlk + 2

∂r

∂xk

∂r

∂xl

−(1 − 2ν)

(∂r

∂xlnk − ∂r

∂xknl

)](43)


whereG is the shear modulus,ν is Poisson’s ratio,nl andnk are the direction cosines of the normal with respect toxlandxk, r is the distance vector between any point(xl, xk)

in the domain and the source point ‘i’ where the unit load isapplied, and∂r/∂n is the derivative of the distance vectorr

with respect to the normal (refer toFig. 1).In order to solve the integral equation numerically, the

boundary is discretized into a series of elements over whichdisplacements and tractions are written in terms of theirvalues at a series of nodal points. In the discrete form, theboundary integralEq. (42)may be written as:

ciwi=∑m

∫S(m)

ω∗NT dS

tm−

∑m

∫S(m)

t∗NT dS

ωm

(44)

whereN are the shape functions for the boundary elements.Typical boundary conditions to be applied in the solution

of Eq. (44)consist of: (a) imposed tractions along the arc ofcontact between the roll and the workpiece, as the workpieceinduces compressive stresses in the rolls; and (b) imposeddisplacements at the contour of the roll neck.

2.4. Coupling between the workpiece and the rolls

The coupling between the workpiece and the driven rollsis achieved by combining the element free Galerkin analy-sis of the deforming workpiece with the boundary elementelastic analysis of the rolls. In other words, the EFG contactpressure acting along roll-workpiece interface is applied asinput data of the BE analysis of the rolls (please refer to arcCD of Fig. 1).

The rolls are discretized with constant type boundary ele-ments. The contour of each roll is subdivided into two differ-

Fig. 2. Theoretical distribution of the horizontal (left) and vertical (right) elastic displacements (units in mm) of the roll during steady-state element freeGalerkin simulation of the rolling of steel with 32% reduction (case 4 ofTable 1—discretization of the domain by 21× 3 nodes).

ent parts; the external contour,lb1, and the internal contour,lb2. The later corresponds to the neck.

In the present investigation the diameter of the neck wasassumed to be equal to 40% of the outside diameter of theroll. The boundary conditions utilised in the linear elasto-static analysis of the rolls were taken as follows,

free contour oflb1 : t = 0

arc of contact(BC)of lb1 : t = −σn

neck(lb2) : ω = 0

(45)

During the elastic analysis of the roll, its position is moveddownward or upward in order to guarantee the final thicknessof the workpiece at the exit of the plastic deformation region.The coordinates of the nodal pointsxB andxC placed at theintersection between the roll-workpiece contact interface andthe inlet and outlet surfaces bounding the plastic deformationregion are also shifted. From the new positions of these nodalpoints, it is possible to compute a new equivalent roll radiusin the form,

R = L

(h0 − h1)+ (h0 − h1)

4, L = xC − xB (46)

where h0, h1 are the initial and final thicknesses of theslab respectively, andL is the horizontal distance betweenthe entry and the exit positions of the slab during rol-ling.

Since the above-mentioned procedure is to be per-formed iteratively, two simultaneous convergence criteriaare utilised. One measures the error norm of the velocitiesin the rolled workpiece,

η = |$v||v| |v| =

√vTv (47)


Fig. 3. Illustration of roll flattening during steady-state element free Galerkin simulation of the rolling of steel with 32% reduction (case 4 ofTable 1).

and requires such error norm to decrease from iteration toiteration. The second criterion requires the error norm ofthe tractions applied at the arc of contact between the rolland the workpiece to decrease. In general terms, the firstcriterion controls the plastic deformation of the workpiecewhilst the second is utilised for determining the final shapeof the flattened rolls.

3. Results and discussion

To test the validity of the combined element freeGalerkin–boundary element approach for plane strainrolling presented in the previous section, results obtainedfrom the theoretical model are compared with available ex-perimental data[19]. Geometry and mechanical propertiesutilised in the rolling analysis are summarised inTable 1.

Fig. 4. Theoretical distribution of the non-dimensional vertical stress,σy/σ0 (left), and of the non-dimensional shear stress,τxy/σ0 (right), applied in theroll during steady-state element free Galerkin simulation of the rolling of steel with 32% reduction (case 4 ofTable 1).

Friction along the roll-workpiece contact interface is mod-elled through the utilisation of a friction factormf = 0.2[19]. It is further assumed that this value remains constantthroughout the deformation.

3.1. Elastic analysis of the roll

Fig. 2 plots the horizontal and vertical deflection of theroll obtained from the linear elastostatics boundary elementanalysis. As it can be seen, the deformation is mainly con-centrated at the vicinity of the arc of contact between theroll and the workpiece. The displacement in thex-directionpresents positive values at the back-slip region and negativevalues at the front-slip region, i.e. near the outlet surface.The detail shown inFig. 3 presents a comparison betweenthe original (rigid) and the flattened (non-rigid) contour ofthe roll along the contact interface (BC).


Table 1Geometrical data and mechanical properties used for computations[19]

Case h0 (mm) h1 (mm) Reduction (%)

1 0.50 0.46 82 0.50 0.42 163 0.50 0.38 244 0.50 0.34 32

Radius of the rollsR = 65 mm, linear velocity of the rollsU = 0.25 m/s.Material: Steelσ = 358(1 + ε/0.044)0.3 N/mm2.

Regarding the distribution of the normal and tangentialstresses shown inFig. 4, higher values ofσy are found tooccur at the roll-workpiece contact interface as well as in theregion of the neck facing it. As expected the vertical stressincreases in the neighbourhood of the neutral point and de-creases monotonically towards the entry and exit boundaries.The distribution of tangential stressτxy discloses the varia-tion of the friction-induced shear stresses applied in the in-coming workpiece along the arc of contact, and reveals theposition of the neutral point.

3.2. Rolling torque and separating force

The comparisons between the theoretical predictions andthe experimental values for the rolling torque and separatingforce as a function of the rolling reduction are shown inFigs. 5 and 6. Two different numerical procedures were takeninto account.

Firstly, the rolls were assumed as rigid in order to re-produce the previous investigations performed by the au-thors[18]. As it can be seen, the theoretical predictions forthe mean separating force and torque under the assumptionof rigid rolls are generally underestimated. Similar resultshad also been obtained by Li and Kobayashi[25] using therigid-plastic finite element analysis.

The second numerical procedure incorporates the elasticdeflection of the rolls through the proposed EFG–BEM ap-proach. The corresponding results are also depicted inFigs. 5

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40 45

Reduction %

To

rqu

e k

N m

/m

Experimental (Shida and Awazuhara)

Theoretical (Rigid roll)

Theoretical (Non-rigid rolll)

Fig. 5. Comparison of measured and computed roll torque (per unit ofwidth) as a function of the percentage of reduction.

0

1000

2000

3000

4000

5000

0 5 10 15 20 25 30 35 40 45

Reduction %

Ro

ll F

orc

e k

N/m

Experimental (Shida and Awazuhara)

Theoretical (Rigid roll)

Theoretical (Non-rigid roll)

Fig. 6. Comparison of the measured and computed roll separating force(per unit of width) as a function of the percentage of reduction.

and 6(please refer to the dashed lines) and confirm a signif-icant improvement of the overall estimate of the mean sep-arating force and torque against the first theoretical resultsobtained under the assumption of rigid rolls.

4. Conclusions

In the present investigation, plane strain rolling is mod-elled by combining the element free Galerkin method withthe boundary element method. The major innovation of thiswork is the development of a new methodology for calcu-lating the deformation of the rolls, based on a linear elasto-static boundary element formulation.

The new proposed methodology offers important compu-tational advantages over classical numerical solutions basedon a full discretization of both the workpiece and the rolls.The main advantage is the possibility of modelling only thecontour of the rolls, leaving the calculation of the field vari-ables inside the rolls as a task to be performed at the post-processing stage. As a consequence, the approach is foundto greatly reduce the overall size of the numerical model.

A comprehensive examination of the proposed method-ology was performed under the numerical and experimentalanalysis of the flat rolling of steel. It has been found that,there were significant improvements in the theoretical pre-dictions of the rolling torque and roll separating force whenthe elastic deflection of the rolls was taken into account.

Acknowledgements

Dr. Xiong is grateful to the Fundação para a Ciencia e aTecnologia de Portugal for the postdoctoral research fellowgrant.


References

[1] L.B. Lucy, A numerical approach to the testing of the fission hy-pothesis, Astron. J. 82 (1977) 1013.

[2] R.A. Gingold, J.J. Monaghan, Smoothed Particle hydrodynamics:theory and application to non-spherical stars, Mon. Not. R. Astron.Soc. 181 (1977) 375.

[3] B. Nayroles, G. Touzot, P. Villon, Generalizing the finite elementmethod: diffusive approximation and diffuse elements, Comput.Mech. 10 (1992) 307.

[4] T. Belytschko, Y.Y. Lu, L. Gu, Element free Galerkin methods, Int.J. Numer. Methods Eng. 37 (1994) 229.

[5] W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods,Int. J. Numer. Methods Eng. 20 (1995) 1081.

[6] C.A. Duarte, J.T. Oden, An h-p adaptive method using clouds, Com-put. Methods Appl. Mech. Eng. 139 (1996) 237.

[7] J.M. Melenk, I. Babuska, The partition of unity finite element method:basic theory and applications, Comput. Methods Appl. Mech. Eng.139 (1996) 289.

[8] E. Onate, S. Idelson, O.C. Zienkiewicz, R.L. Taylor, A stabilizedfinite point method for analysis of fluid mechanics problems, Comput.Methods Appl. Mech. Eng. 139 (1996) 315.

[9] W.K. Liu, Y. Chen, S. Jun, T. Belytschko, C. Pan, R.A. Uras, C.T.Chang, Overview and applications of the reproducing kernel particlemethods, Arch. Comput. Methods Eng.: State Art Rev. 3 (1996) 3.

[10] G. Yagawa, T. Furukawa, Recent developments of free mesh method,Int. J. Numer. Methods Eng. 47 (2000) 1419.

[11] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Mesh-less methods: an overview and recent developments, Comput. Meth-ods Appl. Mech. Eng. 139 (1996) 3.

[12] S.F. Li, W.K. Liu, Meshfree and particle methods and their applica-tions, Appl. Mech. Rev. 55 (2002) 1.

[13] J.S. Chen, C. Pan, C.T. Wu, C. Roque, A Lagrangian reproducingkernel particle method for metal forming analysis, Comp. Mech. 21(1998) 289.

[14] J.S. Chen, C. Roque, C. Pan, S.T. Button, Analysis of metal formingprocess based on meshless method, J. Mater. Proc. Technol. 80(1998) 642.

[15] J.S. Chen, H.P. Wang, W.K. Liu, Meshfree method with enhancedboundary condition treatment for metal forming simulation, in: Pro-ceedings of the 1999 NSF Design and Manufacturing Grantees Con-ference, Queen Mary, USA, 1999.

[16] S. Kulasegaram, Development of particle based meshless method withapplications in metal forming simulations, Ph.D. Thesis, Universityof Wales, Swansea, 1999.

[17] J. Bonet, S. Kulasegaram, Correction and stabilization of smoothparticle hydrodynamics methods with applications in metal form-ing simulations, Int. J. Numer. Methods Eng. 47 (2000)1189.

[18] X. Shangwu, J.M.C. Rodrigues, P.A.F. Martins, Simulation of planestrain rolling using the element free Galerkin method, in: Proceedingsof the 2nd ICFG Workshop on Process Simulation in Metal FormingIndustry, Padova, Italy, 2002.

[19] S. Shida, H. Awazuhara, Rolling load and torque in cold rolling, J.Jpn. Soc. Tech. Plast. 14 (1973) 267.

[20] K. Osakada, J. Nakano, K. Mori, Finite element method forrigid-plastic analysis of metal forming—formulation for finite defor-mation, Int. J. Mech. Sci. 24 (1982) 459.

[21] C. Chen, S. Kobayashi, Rigid plastic finite element analysis ofring compression, Applications of numerical methods to formingprocesses, ASME-AMD 28 (1978) 163.

[22] X.H. Liu, Experiments and analysis by rigid-plastic FEM on theprocess of rolling H-beam with tensions on a Universal Mill (inChinese), Ph.D. Thesis, North-Eastern University, China, 1985.

[23] K. Mori, M. Oyane, K. Osakada, Some problems and its treat-ment techniques in rigid-plastic finite element method—research onrigid-plastic finite element method, J. Jpn. Soc. Tech. Plast. 21 (1980)593.

[24] C.A. Brebbia, J. Dominguez, Boundary Elements—An IntroductoryCourse, McGraw Hill, 1989.

[25] G.J. Li, S. Kobayashi, Rigid plastic finite element analysis of planestrain rolling, J. Eng. Ind. 104 (1982) 55.

[26] X. Shangwu, W.K. Liu, J. Cao, J.M.C. Rodrigues, P.A.F. Martins,On the utilization of the reproducing kernel particle method forthe numerical simulation of plane strain rolling, Int. J. Mach. Tool.Manu. 43 (2003) 89.

Simulation of Plane Strain Rolling Through a Combined Element Free Galerkin-Boundary Element...

Documents

Transcript of Simulation of Plane Strain Rolling Through a Combined Element Free Galerkin-Boundary Element...