Modeling of Casting, Welding and Advanced Solidification Processes XI Edited by M. Schlesinger TMS (The Minerals, Metals & Materials Society), 2006

COUPLED THERMOMECHANICAL SIMULATION OF

SOLIDIFICATION AND COOLING PHASES IN CASTING PROCESSES

M. Chiumenti, M. Cervera and C. Agelet de Saracibar

1E.T.S. Ingenieros de Caminos, Canales y Puertos Universidad Politécnica de Cataluña UPC

C/Gran Capitán s/n, Modulo C1, 08034 Barcelona, Spain

Keywords: Thermo Mechanical simulation, FE analysis, Casting, Solidification

Abstract This paper presents an up-to-date finite element numerical tool for the simulation of aluminium foundry processes. A fully coupled thermo-mechanical formulation including phase change phenomena is considered. The mathematical framework to account for both thermal and mechanical constitutive and boundary assumptions is introduced. The proposed constitutive model is consistently derived from a thermo-elasto-viscoplastic free energy function. Mechanical and thermal material properties are assumed to be temperature-dependent. A continuum transition from the initial fluid-like to the finial solid-like behavior of the part is modeled considering a temperature dependent viscoplastic-surface evolution. Phase-change contribution is taken into account assuming both latent-heat release and shrinkage effects. An accurate definition of the interfacial heat transfer between the solidifying casting and the mold is introduced as an essential ingredient to produce a reliable casting model. In fact, both the solidification process and the temperature evolution strongly depend on the heat exchange at the contact interface. This exchange is affected by the insulating effects of the air-gap due to the thermal shrinkage of the casting part during the solidification and cooling phases. The need for a so closely coupled formulation is the reason why the finite element code VULCAN, developed by the authors, is presented as an accurate, efficient and robust numerical tool, allowing the numerical simulation of solidification and cooling processes for the casting industry.

Introduction: need for a coupled thermo-mechanical formulation The aim of this work is to show the necessity of a coupled thermo-mechanical analysis for the simulation of foundry processes. Up to now, mostly purely thermal simulation has been considered to study the evolution of the solidification and cooling phenomena. This is mainly due to the fact that a (purely) thermal analysis is easier and less costly, and therefore more convenient for large-scale industrial simulations. Moreover, it must be pointed out that in the case of sand die casting thermal results are not so much affected by the mechanical behavior due to the low conductivity and stiffness of the sand. However, an accurate modeling of stresses and deformations during the solidification and cooling phases of the part is essential to capture the accurate thermal pattern in aluminium casting or, more generally, when a metallic mold is used. In fact, the thermal deformation of both part and mold modify the original interfacial heat transfer among all the casting tools involved in the process. The relationship between heat transfer coefficients and air-gap has been closely observed.

Thermal solution The local governing equation that controls the thermal problem, i.e. the temperature and solidification evolution, is stated as a function of an enthalpy state variable H as:

DRH ++⋅−∇= q& (3)

where the thermo-mechanical dissipation can be split into mechanical dissipation mechD and thermal dissipation therD as 0≥+= thermech DDD , respectively defined as:

00

≥=≥=

Θε:σε:σ&&

ther

vpmech

DD (4)

where Θε and vpε are the thermal and viscoplastic components of the total strain tensor ε , respectively. It must be pointed out that both terms are usually negligible in case of casting simulations if compared with the much higher heat flux generated by the thermal gradient between the part and the mold.

Enthalpy rate H& can be defined as a function of the temperature field Θ , as:

( ) ( )Θ+Θ=Θ LCH &&& (5) where ( )Θ= CC is the (temperature dependent) heat capacity coefficient and ( )ΘL& is the rate of latent heat released during the solidification process.

Introducing ( )ΘLf as the liquid fraction function a convenient form to compute ( )ΘL& is the following:

( ) ( )ΘΘΘ

=Θ &&d

dfLL L (6)

where L is the total amount of latent heat to be released during the phase-change [1],[3].

The heat flux q is computed as a function of the temperature field through the Fourier’s law as:

Θ∇−= kq (7)

where ( )Θ= kk is the (temperature dependent) heat conductivity. Let Ω be the integration domain with smooth boundaries Ω∂ . Let δϑ be the test function associated to the temperature field Θ . Denoting by ⋅⋅, the inner product in ( )Ω2L , the weak form associated to the energy balance equation takes the following expression [1]:

cradconvcond

qqqqRkLCΩ∂Ω∂

++−−=∇Θ∇++Θ δϑδϑδϑϑδϑ ,,,,,&& (8)

where q is the prescribed normal heat flux on the boundary and radconvcond qqq ,, are the conduction, convection and radiation heat fluxes at the contact interfaces of the casting tools.

The last term of the weak form defined above is probably the most important one, and drives the solidification and cooling evolution. It is possible to observe either experimentally [8] or numerically [3] that a reliable solidification model highly depends on the appropriate definition of the thermo-mechanical heat transfer at the contact interface.

Starting from the radiation heat flux, and assuming that only small deformations of the casting tools can occur, radq can be computed as a direct function of the surface temperatures cΘ and

mΘ of the two bodies in contact and their emissivities cε and mε , as:

( )( )11144

−+Θ−Θ

=mc

mcaradq εε

σ (9)

On the other hand, heat conduction through the contact surface, condq , can be assumed to be proportional to the thermal gap cmg Θ−Θ=Θ between the contact surfaces, in the form:

Θ= ghq condcond (10)

In this case the surfaces of the two bodies are in contact that is no macroscopic air-gap is formed due to the thermal shrinkage of the casting during the cooling phase. As a consequence, the model assumes that a thermal resistance, condR , only arises as a result of the air (gasses) trapped between the mold and the casting surfaces, due to the roughness values measured on those surfaces. In addition, the thermal resistance due to the mold coating can be also considered. As a result, the total thermal resistance condR , can be computed as [8]:

c

c

a

zcond kk

RRδ

+= 5.0 (11)

where 2,2, moldzcastzz RRR += is the mean peak-to-valley height of the rough surfaces, cδ is the

effective thickness of the coating and ak and ck are the thermal conductivity of the gas trapped and the coating, respectively. Moreover, it is also possible to assume that the microscopic interaction between the contact surfaces depends on the normal contact pressure, Nt , so that the heat conduction coefficient, condh , can be defined using the following expression:

( )n

e

N

condNcond H

tR

th ⎟⎟⎠

⎞⎜⎜⎝

⎛=

1 (12)

where eH is the Vickers hardness and 0.16.0 ≤≤ n a constant exponent [1].

Finally, heat convection between the two bodies arises when they separate from each other due to thermal shrinkage. Heat convection convq has been assumed to be a function of coefficient

convh depending on the air-gap, Ng , multiplied by the thermal gap Θg , in the form [9]:

( ) Θ= gghq Nconvconv (13) In this case, the heat transfer coefficient convh is defined according to the thermal resistances due to the air-gap, limited by the value of the conduction heat transfer coefficient:

⎟⎟⎠

⎞⎜⎜⎝

⎛= conv

N

aconv hg

kh ,max (14)

Observe that both heat conduction and heat convection coefficients depend on mechanical quantities such as the contact pressure or the air-gap induced by the actual deformation of the casting tools. As a consequence, the solidification and cooling processes are driven by a heat exchange at the contact interfaces that is non-uniform and that in general cannot be expressed as a direct function of the temperature field. This is the reason why a fully coupled thermo-mechanical simulation is required.

Mechanical partition

The mechanical model for the cast part and the mold material has been formulated to take into account many important features as thermal shrinkage of the cast during the phase-change, a smooth transition from the liquid-like to a solid-like behavior and the incompressibility constraint when the casting is still liquid, among others. To deal with these complex phenomena the mechanical model chosen is based on the recent developments of the authors in the fields of incompressibility in solid mechanics [4], [5]. The mixed variational formulation proposed uses linear displacements and pressure interpolations, leading to robust and flexible triangular or tetrahedral elements suitable for large-scale computation of constrained media problems. An orthogonal sub-grid scale approach [6], [7] is assumed as an attractive alternative to circumvent the Babuska-Brezzi stability condition [2]. The strong format of the balance of momentum equation is stated introducing the hydrostatic pressure p, as an independent unknown, additional to the displacement field, u, as:

⎪⎩

⎪⎨⎧

=−

=+∇+⋅∇

0Kpe

pevol

0bs (15)

where ( )σs dev= is the deviatoric part of the stress tensor σ and ( )Θ= KK is the (temperature dependent) bulk modulus that control the material compressibility. The volumetric part of the elastic deformation evole is defined as:

( ) Θ−⋅∇== etre eevol uε (16) where the thermal deformation Θe is computed taking into account the shrinkage effects during liquid to solid phase-change as follows:

( ) ( ) ( ) ( )

( ) ( )( ) ( )( )[ ]⎪⎪⎩

⎪⎪⎨

⎧

Θ≤ΘΘ−ΘΘ−Θ−ΘΘ=Θ

Θ≤Θ≤Θ−Θ

=Θ∆

=Θ

Θ>Θ

=ΘΘ

SrefSSrefcool

LSS

LS

pcpc

L

ife

iffVVe

if

e

ααρ

ρρ

33ˆ

0

0

(17)

where ( )Θpce and ( )Θcoolê are the thermal shrinkage during phase-change and the thermal deformation during cooling phase, respectively, being ( )Θα the (temperature dependent) dilatation coefficient, 0V the reference volume at the initial pouring temperature and

pcV∆ the total volume change experimentally observed during the phase change. As a result of the stabilized formulation proposed by the authors in [4], the weak form of the balance of momentum equation accounting for the incompressibility behavior is the following:

⎪⎩

⎪⎨

⎧

Π−∇∇=−

++=⋅∇+∇

∑=

Ω∂Ω∂

Nelem

ee

evol

ccs

pqKpqeq

p

1,,,

,,,,,

τ

tvtvbvvsv (18)

where t is the prescribed traction on the boundary, Ω∂ , and ct is the normal pressure at the contact interfaces, CΩ∂ .

Observe that Π , in equation (18), is the smooth projection of the pressure gradient on the finite element space, computed at each time-step as:

p∇=Π ,ww, (19)

It must be pointed out that in case of liquid-like behavior, ∞→K and 0=Θe , so that the second equation in (18) transforms into:

∑=

Π−∇∇=⋅∇Nelem

ee pqq

1,, τu (20)

which is the weak form associated to the equation of incompressibility: 0=⋅∇ u , where a stabilization term has been considered according to the orthogonal sub-grid scale formulation.

Mechanical model: evolution laws The mechanical model for the cast part and the mold material is consistently derived from a thermo-elasto-viscoplastic potential. Constitutive equations for the deviatoric part of the stress tensor s together with the kinematic and isotropic hardening stress-like variables q and q, respectively, are described by the following equations:

( )( )

( ) [ ] ( )[ ]{ }ξδξσσχχ

Hfq

Kf

devG

S

S

vp

+−−Θ−=

Θ−=

−=

∞ exp132

2

0

q

εεs

(21)

where ( )Θ= GG is the (temperature dependent) shear modulus, χK and H are the coefficients of the linear kinematic and isotropic hardening laws and finally, ( )Θ= ∞∞ σσ and ( )Θ= 00 σσ are the (temperature dependent) saturation and initial flow stress parameters.

Coefficient ( ) ( )Θ−=Θ LS ff 1 is the solid fraction function. The visco-plastic strains vpε are derived, together with the evolution laws for the kinematic and isotropic strain-hardening variable χ and ξ , according to the principle of maximum plastic dissipation as:

( )

( )

( )32,,

,,

,,

γγξ

γγχ

γγ

=∂

ΘΦ∂=

−=∂

ΘΦ∂=

=∂

ΘΦ∂=

qq,

q,

q,vp

qs

nq

qsn

sqsε

(22)

where qsqsn

−−

= and ( )nq,1

,,η

γ ΘΦ= qs are the unit normal to the yield surface and the

viscoplastic multiplier, respectively. It is possible to observe that the model considers a temperature dependent J2-yield-surface ( )ΘΦ ,,q,qs defined as: ( ) ( ) 0,,, ≤Θ−−=ΘΦ qRq, qsqs (23)

where ( )Θ,qR is the (temperature dependent) yield-surface radio defined as:

( ) ( ) ( )[ ]qσfqR S −ΘΘ=Θ 032, (24)

It must be pointed out that the yield-surface radio, as well as the hardening effects, gradually reduce as the temperature increase, vanishing when liquidus temperature is reached. As a result,

a purely viscous Norton model is assumed when liquid-like behavior must be simulated. In this case, the deviatoric stress tensor is simply given by:

vpεs &η= (25)

where ( )Θ=ηη is the (temperature dependent) viscosity parameter.

Mechanical contact modeling

There is a wide number of problems where it is necessary to take into account the thermo-mechanical interaction among different bodies in a numerical simulation. High precision of contact behavior is not always required, because the main emphasis can be focused on the overall performance of the structure, as for military impact and crash-worthiness analyses. On the other hand, forming processes such as hot-forging, sheet metal forming or casting analysis demand an appropriate description of the mechanical constraint at the contacting surfaces.

Focusing on foundry processes, the numerical simulation of the mechanical interaction among casting tools such as part, mold, cores, etc... is an extremely important issue to be taken into account. The solidification and following cooling processes as well as de-moulding phase in a casting analysis is a fully coupled problem where either the temperature evolution defines the mechanical behavior or the mechanical interaction at the contact interface affects the heat flux which drive both temperature and solidification evolution. Looking at the definition of heat conduction and heat convection laws it is clear how important is an accurate definition of the contact pressure as well as the prediction of the open air-gaps between casting part and mold surfaces. The relationship between heat transfer coefficients and air-gap has been experimentally proved [4] especially in case of low-pressure as well as high-pressure die-casting processes which use permanent molds, so that a truthful prediction of either contact pressure or air-gap widths is essential to produce a reliable casting model.

In the literature it is possible to find out many different algorithms to study the mechanical contact between deformable bodies such as penalty methods or augmented lagrangian algorithms among others.

It is experience of the authors that both methods presented are easy to be implemented in a non-linear finite element code and their response is generally good if academic benchmarks are considered. Much more tuning work is necessary when a complex thermo-mechanical simulation of a casting process is required. Large penetration are inevitable leading to loss of precision in both contact reactions and air-gap distribution From one side too small values of the penalty parameter leads to an unconstrained problem while on the other side increasing this value the non-linearity of the problem also increases together with both cpu time (ill-conditioning of the iterative solvers) and the number convergence iterations.

The solution adopted in this work considers a strategy usually referred as rigid contact. This algorithm, possibly less interesting from the theoretical point of view, is able to guaranty both accuracy and robustness. The basic idea consists of defining unilateral links between the slave-master pair in the direction of the normal to the contacting surface (local basis). This means that neither fictitious parameter (penalties) nor ill-conditioning are introduced in the analysis. The drawback is a more expensive assembling procedure due to the rotation of the degrees of freedom in both elemental matrices and residual contributions. The non-linearity induced by the contact algorithm disappears even if the loss of quadratic convergence persists due to the contact check-up procedure (link/release dof).

Numerical simulations The formulation presented is illustrated here with a numerical simulation. The goal is to demonstrate the good accuracy properties of the proposed formulation in the framework of infinitesimal strain thermal-plasticity for an industrial aluminium casting analysis. The computations are performed with the finite element code VULCAN developed by the authors, a project supported by the International Center for Numerical Method in Engineering (C.I.M.N.E.). The analysis is concerned with the solidification process of an aluminium (AlSi7Mg) motor block in a steel (X40CrMoV5) mold. Geometrical and material data were provided by Teksid (Aluminium foundry, Italy). Figure-1 shows a view of the finite element mesh used for the part and the cooling system. The full mesh, including the mold, consists in 580.000 tetrahedral P1P1 elements. Aluminium material behavior has been modeled by the fully coupled thermo-viscoplastic model, while the steel mold behavior has been modeled by a simpler thermo-elastic model. The initial temperature is 650ºC for the casting 250ºC for the mold. Cooling system has been kept at 20ºC. The heat transfer coefficient takes into account the air-gap resistance due to the casting shrinkage. Temperature evolution as well as thermal shrinkage during solidification is shown in figures 2. Figure 3a-3b-3c shows temperature, von Mises deviatoric stress distributions and equivalent plastic strains.

Concluding remarks

A formulation for coupled thermo-mechanical problems has been presented. An enthalpy format of the balance of energy equation has been considered to control the latent-heat released during phase-change, leading to the accurate simulation of the temperature evolution during solidification and cooling processes. Stress analysis is essential to define both conduction and convection heat transfer at the contact interfaces. A particular J2 thermoplastic model has been considered. Temperature dependency of both mechanical properties and yield surface allows an to predict deformations and the final residual stresses of the casting tools involved. The model has been successfully applied to the numerical simulation of industrial alluminium foundry processes.

Acknowledgements Support for this work was provided by the European Commission through the Growth-Project GRD1-2000-25243: Shortening Lead Times and Improving Quality by Innovative Upgrading of the Lost Foam Casting Process (FOAMCAST)

References

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3. Cervera, M., Agelet de Saracibar, C. and Chiumenti, M., Thermo-mechanical analysis of industrial solidification processes, Int. Journal for Num. Methods in Engineering, 46 (1999) 1575-1591

4. Chiumenti, M., Valverde, Q., Agelet de Saracibar, C. and Cervera, M. A stabilized formulation for elasticity using linear displacement and pressure interpolations, Comp. Meth. in Appl. Mech. and Eng. (2002) in press.

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7. Codina, R. Stabilized finite element approximation of transient incompressible flows using orthogonal subscales, Comp. Meth. in Appl. Mech. and Eng., 191 (2002) 4295-4321.

Figure 1: Geometry of the aluminium motor-block casting and mould system considered.

Figure 2: Temperature and shrinkage evolution (plane xy)

Figure 3: a) Temperature, b) J2 von Mises distribution, c) Eq. Platic Strains

a) b) c)