constitutive modeling and numerical simulation of casting

ECCM-2001European Conference on

Computational Mechanics

June 26-29, 2001Cracow, Poland

Constitutive Modeling and Numerical Simulation of Casting Materials

M. Chiumenti, C. Agelet de Saracibar, M. Cervera, E. Onate and G. Duffett

Dept. de Resist`encia de Materials i Estructures a l’EnginyeriaUniversitat Politecnica de Catalunya, Barcelona, Spain

e–mail: [email protected]

Key words: Casting simulations, coupled formulation, thermo-mechanical contact

Abstract. The need of increasing productivity, quality and control of casting process requiresthe understanding of the effects of the process variables in the casting model. This is the reasonwhy in the program VULCAN we tried to reproduce via numerical simulations the completecasting process considering filling, solidification and cooling phases. To do that, it was neces-sary to provide a mathematical model that accounts for both thermal and mechanical constitu-tive and boundary assumptions. Moreover an accurate definition of the interfacial heat transferbetween the solidifying casting and the mould was essential in producing a reliable castingmodel. In fact, either solidification or temperature evolution strongly depend on the heat ex-change at the contact interface. This exchange is affected by the insulating effect of the air-gapdue to the thermal shrinkage of the casting part during solidification and cooling phases. Theaim of this work is to show the advances in the simulation of casting processes providing thedetailed formulation of the constitutive equations and evolution laws assumed for the castingmaterial taking into account the transformations in the material behavior during the solidifica-tion according to the phase changes.


1 Introduction

The solution of coupled problems considered consists of an up-to-date finite element numericalmodel for fully coupled thermomechanical systems, focusing in the simulation of solidifica-tion processes of industrial metal parts. The formulation of the governing equations is con-sistently derived within a thermodynamic context. The proposed constitutive model is definedby a thermo-viscoplastic free energy function which includes contribution forphase change.A continuous transition between the initial fluid-like and the final solid-like behavior of thepart is modelled by considering an improved J2-thermo-viscoplastic model. Thus, an elasto-viscoplastic response suitable for solid-like behavior degenerates into a purely viscous modelaccording to the solid fraction function capturing the liquid-like behavior. Either nonlinear kine-matic and isotropic hardening due to plastic deformation or thermal softening of the yield stressdue to the temperature evolution is assumed. Phase change contribution is taken into accountassuming both latent heat release and shrinkage effects during phase change. Fractional stepmethod arising from an operator split of the governing differential equations is considered.Within the time discrete setting, the additive operator splits lead to aproduct formula algorithmand to astaggered solution scheme of the coupled problem.

2 Local governing equations

The local system of partial differential equations governing the coupled thermo-mechanicalproblem is defined by the momentum and energy balance equations, restricted by the inequal-ities arising from the second law of thermodynamics. This system is supplemented by suitableconstitutive equations and prescribed boundary and initial conditions.

2.1 General form of the local balance laws

Let 2 Redim be the set with smooth boundary@ of a continuum body� in the spacedimension Redim. Let [0; T ] be the time interval of interest.

The local form of thebalance of momentum equation also known asCauchy’s equation ofmotion is given by

r � � + b = �odv

dt(1)

where� is the Cauchy stress tensor,r � (Æ) is the reference divergence operator,b is the vectorof forces per unit of volume,�o is the density in the reference configuration andv is the velocityfield.

Thebalance of energy equation can be written as

_E = � : _" �r �Q+R (2)

so that the increase of the internal energy_E per unit of volume consists of three parts: the stresspower� : _" which represents the mechanical work done by the external forces not convertedinto kinetic energy, the heat supplied by the internal sources per unit of volumeR, and the term

2

ECCM-2001, Cracow, Poland

�r �Q which is the heat provided by the flow of thermal energy through the boundary into thesystem. The balance of energy equation is the local form for theFirst law of Thermodynamics.The Second law of Thermodynamics limits the direction of the energy transformations and itpostulates that there exists a state function called entropyS so that

D = � _S � R +� r �

�Q

�

�� 0 (3)

whereD � 0 is a thermomechanical variable usually referred asthermomechanical dissipation.

A stronger format for the previous equation(3) is given by the following relations

Dint = � _S � R +r �Q �0 (4)

Dcond = �Qr�

�� 0 (5)

whereDint andDcond are the internal dissipation and the dissipation by conduction, respec-tively. Equation(4) is known in the literature as theClausius-Plank equation. Taking into ac-count the balance of energy equation(2) the following format for the Clausius-Plank equationis also available

Dint = � _S � _E + � : _" � 0 (6)

Thus, the first order system of local equations that govern the coupled thermo-mechanical prob-lem is the following

_u = v (7)

�o _v = r � � + b (8)

� _S = R�r �Q+Dint (9)

restricted by the inequalities arising from the second law of thermodynamics

D = Dint +Dcond � 0 (10)

Dint = � _S � _E + � : _" � 0 (11)

Dcond = �Qr�

�� 0 (12)

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2.2 Additive decomposition

Let’s assume anadditive decomposition of the total strain tensor" into its elastic andinelasticparts"E and"I , respectively, that is

" = "E + "I (13)

where the elastic strain tensor"E is given by the sum of the effective elastic deformation plusthe thermal deformation,"e and"� , respectively, so that

"E = "e + "� (14)

while the inelastic strain tensor takes care of viscoplastic effects

"I = "vp (15)

Another hypothesis is the split of total entropyS into its elastic andinelastic part,S E andSI ,respectively, as

S = SE + SI (16)

It is now possible to write in a new format the energy equation(4) using the elastic entropySE

as the state variable

� _SE = R �r �Q+Dmech (17)

where the decomposition of the internal dissipationDint into mechanical and thermal part,Dmech andDther, respectively, is assumed

Dint = Dmech +Dther � 0 (18)

so that the thermal dissipation results in

Dther = � _SI� 0 (19)

Taking care of previous assumptions, the first order system of local equations that governs thecoupled thermo-mechanical problem is the following [6], [1], [2]

_u = v (20)

�o _v = r � � + b (21)

� _SE = R�r �Q+Dmech (22)

restricted by the following inequalities

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D = Dmech +Dther +Dcond � 0 (23)

Dmech = � _SE� _E + � : _" � 0 (24)

Dther = � _SI (25)

Dcond = �Qr�

�� 0 (26)

together with the boundary conditions

u = �u on @u � [0; T ]

� � n = �t on @t � [0; T ]

� = �� on @� � [0; T ]

Q � n = qn on @q � [0; T ]

(27)

and the initial conditionsu = uo in � [0]

v = vo in � [0]

� = �o in � [0]

(28)

3 Constitutive equations

Let �"E; �I ;�

�be theHelmholtz free energy function (per unit reference volume) obtained

from the internal energyE�SE; "E; �I

�by the Legendre transform as

= E � �SE (29)

where�I is a set of generic internal variable that defines the material behavior. If we computethe rate of the free energy function and we make use of equation(24) it is possible to write

_ = � : _"� _�SE�Dmech (30)

If we differentiate the free energy function with respect to the state variables we obtain

_ =@

@ "E: _"E+

@

@ �I: _�I +

@

@�_�

=@

@ "E: _"�

@

@ "E: _"I+

@

@ �I: _�I +

@

@�_� (31)

From equations(30) and(31) it is possible to obtain the following inequality

Dmech =

��

@

@ "E

�: _"�

�SE +

@

@ �

�_�+

@

@ "E: _"I�

@

@ �I: _�I

� 0 (32)

Applying Coleman’s method, we obtain the definition of the constitutive equations as

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� =@

�"E; �I;�

�@ "E

(33)

SE = �@

�"E; �I;�

�@�

(34)

where the internal dissipation is given by

Dmech = � : _"I+�I : _�I� 0 (35)

being for definition

�I = �@

�"E; �I;�

�@ �I

(36)

Equation(35) is known in the literature asthe reduced equation of dissipation.

3.1 Definition of the free energy function

We have seen how it is possible to formulate and obtain the constitutive equation that governthe thermo-mechanical problem starting from the definition of the potential

�"E;�; �I

�. A

possible choice of the set of internal variables to particularize the free energy function to thecase of thermo-viscoplastic behavior is given by�I = [�; �], that is the isotropic and kinematichardening variables, respectively. Thus, a possible format of the free energy function =

�"E;�; �;�

�is given as the sum of following contributions

= �"E;�; �;�

�= W

�"E�+ T (�) +M

�"E;�

�+K (�;�) (37)

whereW�"E�, M

�"E;�

�, T (�) andK (�;�) are the elastic stored energy, the coupling po-

tential, the thermal potential including phase change contribution and the plastic hardeningpotential, respectively. The expressions chosen here for these terms are the following

W�"E�=

1

2k tr2

�"E�+G dev2

�"E�

(38)

M�"E;�

�= �k e� (�) "E : 1 (39)

T (�) = �

Z �

�o

d��

Z ��

�o

Cv + dL/ d�

�d� (40)

K (�;�) = fS [�1 � �o]

��

1� exp (�Æ�)

Æ

�

+1

2fS H�2+

1

3fS K k�k

2 (41)

wherek is the bulkG the shear modulus,� the thermal volumetric-change coefficient,�o theinitial temperature field,Cv the heat capacity at constant volume (not including the phase changecontribution),�o the initial flow stress,�1 the saturation hardening limit,H the linear isotropic

6


hardening coefficient,K the linear kinematic hardening coefficient, and finally1 =Æijei ej

the rank-two symmetric unit tensor.

Note that thermal potentialT (�) incorporates the contribution coming from the phase change.This contribution concerns the latent heat released in the temperature range between the liquidusand solidus temperature that can be defined as the product between the total amount of latentheatL and a particular functionfS that controls the heat flow during the phase change process,so that

L (�) = L fS (�) (42)

L (�) is the so calledlatent heat function. In case of a single phase changefS (�) representsthe solid-fraction function so that the latent heat will be released (or absorbed) depending onthe fraction of solid existing in the considered volume. This function takes the form

fS (�) =

8<:

0 if � � �L

0 � fS (�) � 1 if �S < � < �L

1 if � � �S

(43)

where�L and�S are the liquid and solid temperature, respectively.

Terme� (�) is the volumetric thermal deformation and it is defined as

e� (�) =

8<:

0 if � � �L

epc (�) if �S < � < �Lbe� (�) if � � �S

(44)

whereepc (�) is the thermal shrinkage during phase change whilebe� (�) is the thermal defor-mation during cooling phase, respectively defined as

epc (�) =�V

VofS (�) (45)

be� (�) = 3 [� (�) (�� ref)� � (�S) (�S ��ref)] (46)

beingVo is the volume at the initial temperature and�V the total volume change observed dur-ing the phase change;�ref is the environment temperature during the experimental evaluationof the dilatation coefficient� (�).

This given, the constitutive equations that govern the coupled problem result in

� =@

@ "E= k

�tr�"E�� e� (�)

�1+ 2G dev

�"E�

= k ee1+ 2G dev ("e) (47)

SE = �@

@ �=

Z �

�o

Cv

�d��M� +

Z �

�o

dL/ d�

�d� (48)

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It is possible to identify two different contributions to the elastic entropy function, respectivelygiven by

Se =

Z �

�o

Cv

�d��M� (49)

Spc =

Z �

�o

dL/ d�

�d� (50)

Thus, the total entropy function is now given by the sum of following contributions

S = SE + SI = Se + Spc + SI (51)

The mechanical dissipation is obtained as

Dmech = � : _"vp+q _� + q : _� � 0 (52)

where the state variablesq andq are defined as the derivatives of the free energy function withrespect to� and�, respectively, so that

q = �@

@ �= �fS [�1 � �o] [1� exp (�Æ�)]� fS H � (53)

q = �@

@ �= �

2

3fS K� (54)

Finally, let us introduce in this section theFourier’s law as the constitutive law that governs theheat flux, so that

Q = �k (�)r� (55)

wherek (�)=k (�) 1 is the conductivity tensor.

4 Evolution laws

The basic idea is to try to formulate a model the closest as possible to the experimental ob-servation. The main difficulty is the very different behavior of the liquid phase compared withthe solid phase. The liquid phase is characterized by a purely viscous behavior so that both theelastic and the plastic deformation must be neglected. On the other hand, when the material issolid a standard elastic-viscoplastic behavior should be taken into account. Let us consider avisco-plastic potential� (s;q;q;�) to deal with von Mises yield criterion with an isotropic andkinematic hardening combined with a thermal softening effect, defined as

� (s;q;q;�) = ks� qk � R (q;�) � 0 (56)

8


wheres =dev (�) is the deviatoric part of the stress tensor andR (q;�) is the radius of the yieldsurface given by

R (q;�) = fS (�)

r2

3[�0 (�)� q] (57)

According to the principle of maximum plastic dissipation, the evolution laws for the plasticvariables can be obtained as

_"vp = @ � (s;q;q;�)

@ s= n (58)

_� = @ � (s;q;q;�)

@ q= � n (59)

:

� = @ � (s;q;q;�)

@ q=

r2

3(60)

wheren is the unit normal to the yield surface and is the viscoplastic parameter, respectivelygiven by

n =s� q

ks� qk(61)

=1

� (�)h� (s;q;q;�)i

n (62)

being� (�) andn the visco-plastic parameters associated to the model.

Note that if� � �L the viscoplastic potential� (s;q;�) degenerate in a purely-viscous poten-tial

� (s;q;q)! � (s) = ksk (63)

The evolution of the plastic multiplier transforms into

=1

� (�)h� (s)i ! =

1

� (�)ksk (64)

Therefore, it is possible to obtain a purely viscous relation between the deviator of the stresstensors and the rate of visco-plastic strain_"vp, governed by the viscous parameter�

s =� (�) _"vp (65)

In this condition, if the value of the viscous parameter� is enough small compared with theshear modulusG, then the total deformation will be essentially visco-plastic, to say irreversiblevery close to the experimental observation.

9


Finally, observe that the evolution law of the inelatic entropy variableS I the can be obtainedby analogy as

_SI = vp@�

@�= � vp

r2

3

d �0 (�)

d�

and if we introduce this result into the expression of thermal dissipation(19) then

Dther = � vpr

2

3

d �0 (�)

d�� 0 (66)

leading to the restrictiond �0 (�)

d�� 0 (67)

that is, onlythermal softening is allowed.

5 Equivalent forms of the energy equation

Starting from the entropy form of the balance of energy equation given by

� _S = R�r �Q+Dint (68)

and taking into account the additive decomposition of the entropy function(51) it is also possi-ble to consider the following equivalent system of equations

� _Se = R�r �Q+Dmech �_L (69)

� _Spc = _L (70)

� _SI = Dther (71)

showing the local evolution of the different contributions of the entropy function. Applying thechain rule to the elastic entropy function then it results

� _Se = C (�) _� +Hep (72)

thus, equation(69) can be written as

C (�) _� = R �r �Q+Dmech �_L�Hep (73)

where the following notation has been introduced

C (�) = �@Se

@�= Cv � �M�� (74)

Hep = �

�@Se

@"E: _"E +

@Se

@�_�

�= ��

�@�

@�: _"E �

@q

@�_�

�(75)

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beingC (�) theheat capacity (not including the phase change contribution) andH ep thestruc-tural elasto-plastic heating [6]. Finally, equation(73) can also be written in entalpy form as

_H = R�r �Q+Dmech �Hep (76)

where the rate of the entalpy functionH is defined as

_H =

�C (�) +

dL (�)

d�

�_�

6 Formulation of the Contact Problem

In this section the main focus is the formulation and treatment of the contact problem recoveringthe same general structure used to describe the constitutive framework for the bulk continua.

6.1 Local contact governing equations

Let us extend the balance equations coming from the first and second laws of thermodynamics,to account for the contributions at the contact interfaces.

First, let us consider the balance of energy equation. If we denote with_E� the time derivativeof the total internal energy given by

_E� =

Z

_E dV +

Z�c

_Ec dS (77)

being _E the stored energy per unit volume in the bulk medium(see eq: 2) and _Ec the storedenergy per unit area on the contact interface�c defined as

_Ec = tN _gN +Qc (78)

wheregN andtN are the normal gap and normal contact pressure, respectively, andQc is theheat flux per unit of contact surface�c supplied at the interface, assumed positive if they areflowing out of the bodies into the interface region.

Let us now consider the second law of thermodynamics. Suppose also in this case that the totalentropy of the systemS� can be given by

_S� =

Z

_S dV +

Z�c

_Sc dS (79)

being _S the entropy per unit volume in the bulk media(see eq: 3) and _Sc the entropy per unitarea on the contact interface�c satisfying the following equation

�c_Sc = Qc (80)

where�c is the contact interface temperature. Introducing this result into equation(78), theenergy equation at the contact interface can be rewritten as

�c_Sc �

_Ec + tN _gN = 0 (81)

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7 Contact Constitutive Equations

In what follows the contact free energy function is introduced. The constitutive equations ofboth contact pressure as well as the heat flux at the contact inteface are obtained.

7.1 Mechanical contact model

Starting from the definition of the internal energy as a function of

Ec = Ec (gN ; Sc) (82)

it is possible to define a free energy functionc via Legendre transformation as

c = Ec � �c Sc = c (gN ;�c) (83)

Following a standard argument in thermodynamics, one may time differentiate equation(83) toobtain

_c = _Ec � �c_Sc �

_�c Sc (84)

=@c

@gN_gN +

@c

@�c

_�c (85)

Taking into account the energy equation at the contact interface(81) then

�tN �

@c

@gN

�_gN �

�Sc +

@c

@�c

�_�c = 0 (86)

which must hold for any_gN and _�c. Applying Coleman’s method, it results

tN =@c

@gN(87)

Sc = �@c

@�c

(88)

Let us now particularize the contact free energy function to obtain the standard restrictions atthe contact interface, assuming the following terms [3]

c (gN ;�c) = Wc (gN) + Tc (�c) (89)

beingWc (gN)andTc (�c) are the stored energy due to the bodies interaction at the contactinterface and the thermo-contact potential, respectively given by

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Wc (gN) =1

22N hgNi

2 (90)

Tc (�c) = Cco [(�c ��o)��c log (�c/�o)] (91)

where�o is the initial temperature field,Cco is the heat capacity associated with the contactinterface and2N is the normal penalty parameter that provides apenalty regularization of thecontact constrains [8].

From this definition the constitutive equations that completely specify the thermomechanicalcontact framework result in

tN = 2N hgNi (92)

Sc = Cco log (�c/�o) (93)

Equation(92) defines the normal contact pressure as

tN =

�0 if gN � 0

2N gN if gN > 0(94)

which are the regularization of the contact constraints of impenetrability and non-adhesion,respectively given in Khun-Tucker form as

tN � 0 (95)

gN � 0 (96)

gN tN = 0 (97)

7.2 Thermal contact model

An accurate knowledge of the interfacial heat transfer coefficient between the a solidifyingcasting and a die is essential in producing a reliable solidification model. Hence, to completethe thermomechanical contact model, reliable heat conduction and heat convection laws mustbe considered.

7.2.1 Heat conduction model

Heat conduction through the contact surfaceQc; cond has been assumed to be a function of co-efficienthcond depending on of the normal contact pressuretN , multiplied by the thermal gapg� between the contact surfaces of the form

Qc; cond = hcond (tN) g� (98)

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In this case we assume that no macroscopical air gap is formed due to the thermal shrinkage ofthe casting during cooling phase. The model assumes that a thermal resistancesRcond arise as aresult of the air trapped between the mould and the casting surfaces due to the roughness valuesmeasured on those surfaces. In addition, it must be also considered the thermal resistance dueto the mould coating, so that

Rcond = 0:5Rz

ka+Æc

kc

whereRz =qR2

z;cast +R2z;coat is the mean peak-to-valley height of the rough surfaces,Æc is the

effective thickness of the coating andka andkc are the thermal conductivity of the gas trappedand the coating, respectively.

Moreover, it is also possible to assume that the microscopical interaction between the contactsurfaces depends on the normal contact pressure so that the heat conduction coefficienthcond

could be defined using the following expression

hcond (tN) =1

Rcond

�tN

He

��

(99)

whereHe is the Vickers hardness and� a constant exponent [10].

7.2.2 Heat convection model

Heat convection between the two bodies arises when they separate from each other due to thethermal shrinkage effect. Heat convection fluxQc; conv has been assumed to be a function ofcoefficienthconv depending on the mechanical gapgN multiplied by the thermal gapg� of theform

Q(i)c; conv = hconv (gN) g� (100)

where in this case the heat transfer coefficienthconv is defined via the inverse of the thermalresistances of both the air gap and the coating

hconv =1

max(gN ;Rz)

ka+ Æc

kc

8 Weak Form of the Governing Equations

Let be integration domain with smooth boundaries@. Let Æ� be the test function associatedto the displacement fieldu. The weak form of the balance of momentum equation(1) in thehypothesis of a quasi-static process results in

hÆ�;r � �i+ hÆ�;bi = 0 (101)

whereb is the vector of forces per unit of volume.

14


Applying the divergence theorem to the first term in the above equation yields

hÆ�;r � �i = �hr (Æ�) ; �i+ hÆ�;�ti@ + hÆgN ; tN i�c (102)

where�t = � � n is the prescribed surface traction. Expression forÆgN can be found in [7] andtherefore will not be given here. Substituting into(101) the result is the standard format of theweak form of the balance of momentum equation given by

hrs (Æ�) ; �i = G (103)

whereG is the mechanical work due to the internal forces and the prescribed surface tractions,as

G = hÆ�;bi+ hÆ�;�ti@

Let Æ# be the test function associated to the temperature field�. Let choose the balance ofenergy equation in entalpy form given by equation(76) then the weak form associated resultsin

DÆ#; _H

E= hÆ#; Ri � hÆ#;r �Qi + hÆ#;Dmechi � hÆ#;Hep

i (104)

Integrating by part the conductivity term in the previous equation

hÆ#;r �Qi = hr (Æ#) ; kr�i+ hÆ#; �qi@ + hÆ#c; Qci�c (105)

where�q = Q � n is the flux normal to the boundaries. Substituting the result in(104) then theweak form of the energy equation is the following

DÆ#; _H

E+ hr (Æ#) ; kr�i = G# (106)

whereG# is the thermal work due to the internal sources, the mechanical dissipation and theprescribed heat flux, given by

G# = hÆ#; Ri+ hÆ#;Dmechi � hÆ#;Hepi � hÆ#; �qi@ � hÆ#c; Qci�c (107)

9 Numerical Simulations

The formulation presented is illustrated here with a numerical simulation. The goal is to demon-strate the good accuracy properties of the proposed formulation in the framework of infinitesi-mal strain thermal-plasticity for an industrial solidication example.

The computations are performed with the finite element code VULCAN developed by theauthors, project supported by the International Center for Numerical Method in Engineering(C.I.M.N.E.). Newton-Raphson method, combined with a line-search optimization procedure,is used to solve the nonlinear system of equations arising from the spatial and temporal dis-cretization of the weak form of the governing equations. Convergence of the incremental iter-ative solution procedure was monitored by requiring a tolerance of 0.1% in the residual basederror norm.

15


Figure 1: Solidification of a suspension component: FE mesh used.

The analysis is concerned with the solidification process of an aluminium (AlSi7Mg) specimenin a steel (X40CrMoV5) mould. Geometrical and material data, as well as experimental results,were provided by TEXID. Fig. 1 shows a view of the finite element mesh used, consisting in246.600 tetrahedral elements.

Aluminium material behavior has been modelled by the fully coupled thermo-viscoplasticmodel, while the steel mould behavior has been modelled by an easier thermo-elastic model.The values of the material properties used in the numerical analysis are confidential (TEXIDmaterial database). The initial temperature is650 oC for the part and250 oC for the mould. Theheat transfer coefficient takes into account the air-gap considering the shrinkage of the catingduring the solidification.

The temperature distribution during solidification is shown in Fig. 2. Fig. 3 and Fig. 4 showthe temperature and von Mises deviatoric stress distribution on sections x-y , respectively. Inthese figures it is also possible to observe the mechanical gap between the part an the mouldresponsable of a non-uniform heat flux at the contact interface.

10 Conclusions

The developments presented in this paper demonstrate that the approach advocated can be theproper strategy in the context of thermo-mechanical phase-change analysis. All the formulation

16


Figure 2: Solidification of a suspension component: temperature contour-fill.

presented has been implemented and tested in the context of the finite element method. Thecomputations have been performed with the finite element code VULCAN developed by theauthors, project supported by the International Center for Numerical Method in Engineering(C.I.M.N.E.).

11 Acknowledgement

Support for this work was provided by the European Commission through the Esprit-Project EP-28144: ”Enhanced Design Environment for Industrial casting Processes on Parallel ComputingPrograms”. This support is gratefully acknowleded.

References

[1] Agelet de Saracibar, C., M. Cervera and M. ChiumentiVulcan-2000: A Finite ElementSystem for the Simulation of Casting Processes. Proceedings of V International Confer-ence on Computational Plasticity, COMPLAS-97, Barcelona, Spain, 1997.

17


Figure 3: Solidification of a suspension component: temperature distribution section x-y.

[2] Agelet de Saracibar, C., M. Cervera and M. ChiumentiCurrent Issues in the NumericalSimulation of Casting Processes. Proceedings of VI Symposium on plasticity and itsCurrent Applications, Juneau, Alaska, U.S.A., 1997.

[3] Agelet de Saracibar, C., M. Cervera and M. ChiumentiOn the Constitutive Modeling ofThermoplastic Phase-change Problems. 7th International Symposium on Plasticity andits Current Applications, Cancun, Mexico, January 5-13, 1999.

[4] Armero, F and J.C. Simo [1992]A New Unconditionally Stable Fractional Step Method forNonlinear Coupled Thermomechanical Problems. Int. Journal for Num. Meth. in Engin.,Vol. 35, pp. 737-766.

[5] Armero, F and J.C. Simo [1992]Product Formula Algorithms for Nonlinear CoupledThermoplasticity: Formulation and Nonolinear Stability Analysis. Sudam Report #92-4,Department of Mechanical Engineering, Stanford University, Stanford, California, 1992.

[6] Armero F. [1993] Numerical Analysis of Dissipative Dynamical Systems in Solid andFluid Mechanics, with Special Emphasis on Coupled Problems. Ph.D. Thesis, Departmentof Mechanical Engineering, Stanford University, Stanford, California, # 94305.

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Figure 4: Solidification of a suspension component: Von mises deviatoric stress distribution:section x-y.

[7] Laursen, T.A. and J.C. Simo [1991]On the Formulation and Numerical Treatment ofFinite Deformation Frictional Contact Problems. Nonlinear Computational Mechanics–State of the Art., P. Wriggers & W. Wagner, eds., Springer-Verlag, Berlin, pp. 716-736.

[8] Laursen, T.A. and J.C. Simo [1993]A Continuum-Based Finite Element Formulationfor the Implicit Solution of Multi-Body, Large Deformation Frictional Contact Problems.International Journal for Numerical Methods in Engineering, Vol. 36, pp. 3451-3485.

[9] Ransing, R.S. and R.W. Lewis [1998]A Thermo-Elasto-Visco-Plastic Analysis for De-termining Air Gap and Interfacial Heat Transfer Coupled with the Lewis-Ransing Cor-relation for Optimal Feeding Design. Proceedings of VIII International Conference onModelling of Casting, Welding and Advanced Solidification processes, San Diego, Cali-fornia, U.S.A., 1998.

[10] Zavarise, G., P. Wriggers, E. Stein and B. A. Schrefler [1992]A Numerical Model forThermomechanical Contact Based on Microscopic Interface Law. Mechanics ResearchCommunications Vol. 19(3), pp. 173-182.

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constitutive modeling and numerical simulation of casting

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