Post on 29-May-2018
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
1/36
A large strain thermoviscoplastic formulation
for the solidication of S.G. cast iron in a
green sand mould
Diego J. CelentanoDepartamento de Ingeniera Mecanica, Universidad de Santiago de Chile, Av. Bdo. O'Higgins 3363,
Santiago de Chile, Chile
Received in nal revised form 24 March 2000
Abstract
This paper presents a large strain thermoviscoplastic formulation for the analysis of thesolidication process of spheroidal graphite (S.G.) cast iron in a green sand mould. This for-
mulation includes two dierent non-associate constitutive models in order to describe the
thermomechanical behaviour of each of such materials during the whole process. The perfor-
mance of these models is evaluated in the analysis of a solidication test. # 2001 Elsevier
Science Ltd. All rights reserved.
Keywords: Cast iron and green sand constitutive behaviours; A. Phase transformation; A. Solidication;
A. Thermomechanical processes; B. elastic-viscoplastic material
1. Introduction
Several thermomechanical models to simulate dierent casting processes have
been formulated during the last years (see Zabaras et al., 1990; Inoue and Ju, 1992;
Bellet et al., 1996; Celentano et al., 1996; Trovant and Argyropoulos, 1996 and
references therein). In particular, some of them have been used to analyze the soli-
dication and subsequent cooling of spheroidal graphite (S.G.) cast iron in green
sand moulds considering relatively simple constitutive models for these materials
(Celentano et al., 1995; Agelet de Saracibar et al., 1999). Although they predicted a
satisfactory agreement between experimental and numerical results, it has been long
recognized that the use of more sophisticated models is necessary to represent in amore realistic form some particular physical aspects involved in this process
(Hamata, 1992; Azzouz, 1995; Ami Saada et al., 1996; Celentano, 1997).
International Journal of Plasticity 17 (2001) 16231658
www.elsevier.com/locate/ijplas
0749-6419/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.
P I I : S 0 7 4 9 - 6 4 1 9 ( 0 0 ) 0 0 0 9 5 - 4
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
2/36
In this context, the aim of this paper is to present a large strain thermoviscoplastic
formulation which includes microstructural liquidsolid and solidsolid phase-
change eects, incorporates volumetric expansions due to the metallurgical trans-
formations, deals with temperature-dependent hardening laws and describes thehygrometric and damage phenomena experienced by the sand. In particular, the
thermomechanical formulation is presented in Section 2 while Sections 3 and 4
describe the S.G. cast iron and sand constitutive models, respectively. These
descriptions start with the choice of the viscoplastic and phase-change internal
variables with their respective evolution equations. Afterwards, a specic free energy
function intended to predict the material behaviour during the whole solidication
and cooling process is proposed as a function of the Almansi strain tensor, the
internal variables and the temperature. This denition is an extension, considering
large strains, microstructural phase-changes, mixed response for the dierent phases
and damage eects, of the specic free energy function developed, used and partially
validated with experiments by Celentano et al. (1995) and Celentano (1997). At this
point, the standard procedures in thermodynamics allow to derive all the con-
stitutive equations involved in the formulation, namely the stress-strain law, the
entropy function, the tangent conjugate of the thermal dilatation tensor, the internal
heat source, the conjugate of the internal variables and the expression of the dis-
sipation. As a consequence of this proposed free energy function, it is possible to
dene the elastic contribution of the Almansi strain tensor and thus to recover the
additive decomposition of such tensor which, in turn, it is shown to be consistent
with a particular kinematic decomposition of the deformation gradient tensor. Asreported by Lubliner (1990), this approach is equivalent to other thermomechanical
theories that consider the kinematic decomposition and the specic free energy
function written in terms of the elastic part of a strain measure as starting points to
derive the constitutive equations of the formulation (Wriggers et al., 1989; Armero
and Simo, 1993; Levitas, 1998).
Solidsolid phase-transformations are very important processes that have been
extensively analysed by dierent researchers (see Fischer et al., 1996; Levitas, 1998;
Levitas et al., 1998; Cherkaoui et al., 1998 and references therein) with particular
emphasis to martensitic phase transitions. Some complex aspects of these phenom-
ena, such as transformation-induced plasticity, nucleation criterion, interface propa-gation, nucleus nondissappearance conditions and displacements discontinuities, have
been recently taken into account in the development of dierent thermomechanical
formulations (see e.g. Levitas, 1998). In the present work, however, a more simpler
model is adopted due to the fact that low temperatures rates are considered during the
solid-solid phase-change and, therefore, it can be modelled by means of nucleation
and growth laws together with a transformation-induced plasticity equation already
used by Hamata (1992) and Celentano (1997) for ferritic S.G. cast iron.
The motivation to include large deformation concepts in this proposed formula-
tion is based on the invalidity in a strict sense of the innitesimal strain assumption
made by some existing models (Celentano et al., 1996, 1999) that may occur whendescribing the quasi-incompressible material behaviour in the liquid phase or in the
mushy zone at high temperatures.
1624 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
3/36
Moreover, it should be noted that the consideration of some microstructural
aspects for both the liquidsolid and solidsolid phase-changes in this thermo-
mechanical context in a coupled form is also a relevant feature of the present for-
mulation. This phenomenological approach includes the denition of evolution lawsfor the S.G. cast iron phase-change internal variables that are assumed to describe
the average microstructure formation occurring in a certain volume at the macro-
scopic level avoiding in this form a microscopic scale modelling of the micro-
mechanisms involved in the phase transformation. Hence, the aim of considering
this assumption in the thermomechanical/microstructural formulation proposed in
this work is to describe in a phenomenological way such complex eects in solidi-
cation problems.
Although little rate-sensitiveness of green sand behaviour could be observed dur-
ing the experiments conducted by Azzouz (1995) and Ami Saada et al. (1996) and
taking into account that the rate-independent plasticity can be considered as a par-
ticular case of viscoplasticity (Lubliner, 1990), a thermoviscoplasticity framework
has been chosen for both the S.G. cast iron and sand models in order to simplify
their presentations.
All these features have been implemented in a coupled thermomechanical/micro-
structural nite element model briey described in Section 5. Further, a strain-dis-
placement matrix able to deal with large strain situations avoiding numerical locking
due to the incompressibility of viscoplastic ows is also proposed. Finally, Section 6
presents the analysis of a solidication test where some available experimental
measurements are compared with the numerical results obtained using this proposedformulation.
2. Thermomechanical formulation
In a general thermomechanical context, the existence of a specic Helmholtz free
energy function i i E; k; T can be assumed as a function of the Green-Lagrange strain tensor E, the nnt-dimensiona1 vector eld k 1; F F F ; nntY nnt51 of phenomenological internal state variables k (usually governed by evolution
equations with zero initial conditions) and the temperature T (see Coleman andGurtin, 1967; Lubliner, 1990; Levitas, 1996, 1998 and references therein for this and
other equivalent expressions for the specic free energy function). With this deni-
tion of i, the Second PiolaKirchho stress tensor S can be derived using the
classical constitutive relation (Coleman and Gurtin, 1967): S 0@i=@E, where 0is the density at the material (initial) conguration. As pointed out by Doyle and
Ericksen (1956), an equivalent stress denition is obtained in the form: 0@e=@e(see also Lubliner, 1985 for further details in the derivation of this expression),
where is the Kirchho stress tensor ( FSFT, F being the deformation gradienttensor and T the transpose symbol), e is the Almansi strain tensor and e
e e;F; k;T is the reformulated specic free energy function written in terms ofEulerian arguments. Therefore, the Cauchy stress tensor , dened by =0(Malvern, 1969), can be computed in this framework as @e=@e, where is the
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1625
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
4/36
density at the spatial conguration. Moreover, by considering the assumptions of
small elastic strains and isotropic material response (usually accepted for metals and
other materials; see Murnagaham, 1937; Garcia Garino and Oliver, 1992; Garcia
Garino, 1993 and references therein), the specic free energy function may be writ-ten as e; k;T . This last simplied form of is adopted in the formulation tobe described below.
Taking into account the previous considerations, the existence of a specic
Helmholtz free energy function e; k;T leads to the following form of thegoverning local equations (all of them valid in where is the spatial cong-uration of a body and denotes the time interval of interest with t P ) expressedby the mass conservation, the equation of motion, the energy balance and the dis-
sipation inequality (Malvern, 1969):
J 0 I
r bf u P
cT: rq r T X d rnt 0 Q
qrT Dnt50 R
together with appropriate boundary and initial conditions and the constitutive rela-
tions given by: @=@e is the Cauchy stress tensor, @=@T is the specicentropy function, c T@2=@T2 is the tangent specic heat capacity, q krTisthe heat ux vector dened according to the Fourier law (k being the conductivity
tensor), b @2=@e@T @=@T is the tangent conjugate of the thermal dilata-tion tensor, rnt 10 T@qk=@T qk
Dk=Dt is the specic internal heat source
and Dnt qkDk=Dt is the internal dissipation where qk 0@=@k are the con-jugate variables ofk. According to the nature of each internal variable, the symbols
* and D =Dt appearing in the previous expressions respectively indicate an appro-priate multiplication and a time derivative satisfying the principle of material frame-
indierence (Malvern, 1969). Further, r is the spatial gradient operator, bf is thespecic body force, the superposed dot indicates time derivative, r is the specic heatsource, u is the displacement vector, J> 0 is the determinant of the deformation
gradient tensor F(F1 1 ! u, with 1 being the unity tensor) and d is the rate-of-deformation tensor (d 1=2 r v v r , where v u: is the velocity vector).Instead of Eq. (4), an additional more restrictive dissipative assumption reads: qrT50 and Dnt50 (see Coleman and Gurtin, 1967). The rst condition is auto-matically fullled for a semi-positive denite conductivity tensor (in the isotropic con-
text assumed in this work, this condition leads to a non-negative conductivity
coecient) while the second imposes restrictions over the constitutive model denition.
It is seen that the denition of and consequently of Dk=Dt are essentials fea-
tures of the formulation in order to describe the thermomechanical/microstructuralbehaviour of the materials involved in the solidication process. To this end, the
following split is proposed: nnt nvpnt npcnt, where nvpnt and npcnt refer to the number
1626 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
5/36
of internal variables related to viscoplastic (non-reversible that may occur in every
material) and phase-change (for S.G. cast iron) or hygrometric (for green sand)
eects, respectively. In this context, this assumption leads to rnt
r
vpnt
r
vpnt and
Dnt Dvpnt Dpcnt. The evolution equations for the viscoplastic internal variables aredened within the non associate thermoviscoplasticity theory framework (Perzina,
1971) as Dk=Dt gk where gk k@=@qk (no sum on k) are known functionsdened in terms of a function k and a viscoplastic potential which in turn may
depend (among other variables) on a yield function F F e; k;T k 1; F F F ; nnt such that no viscoplastic evolutions occur when F< 0 (Perzina, 1971; Hamata,
1992; Arnold and Saleeb, 1994; Azzouz, 1995; Celentano, 1997). On the other hand,
the rate equations for the phase-change internal variables of the S.G. cast iron are
given by microstructural models based on kinetics considerations (Banerjee and
Stefanescu, 1991; Chang et al., 1991; Goettsch and Dantzig, 1994; Celentano and
Cruchaga, 1999; Onsien et al., 1999) and the hygrometric internal variable
accounts for the water content existing in the sand (Azzouz, 1995). Moreover, the
specic free energy function is decomposed into a thermoelastic, thermoviscoplastic
and phase-change (for S.G. cast iron) or hygrometric (for green sand) contributions.
The viscoplastic potential is also proposed in additive form for the S.G. cast iron
considering a thermoviscoplastic and phase-change terms while a purely thermo-
viscoplastic behaviour is assumed for the sand (Celentano, 1997). Some details of
the S.G. cast iron and green sand constitutive models are given below.
3. S.G. cast iron model
S.G. cast iron (usually called ductile or nodular iron) has become an engineering
material of major importance since this ternary alloy has many of the mechanical
advantages of steel with the processing economies of cast iron. In this type of cast
iron, the adding of magnesium causes the graphite akes to form in shape of spheres
frequently called spherulites. In spite of decades of study, not all the kinetic
mechanisms involved in the S.G. cast iron solidication are actually known (Flem-
ings, 1974). However, several models have been recently proposed in order to predict
the microstructure formation in castings (see Banerjee and Stefanescu, 1991; Changet al., 1991; Goettsch and Dantzig, 1994; Celentano and Cruchaga, 1999; Onsien et
al., 1999 and references therein).
In a hypoeutectic S.G. cast iron, liquid-solid and solid-solid phase-changes take
place during solidication and cooling. As the temperature of the melt decreases
below the liquidus temperature, primary austenite dendrites are formed. Then, gra-
phite and cementite eutectic grains are developed through nucleation and growth as
the respective eutectic temperatures are reached. The solid-solid or eutectoid trans-
formation is also a competitive growth process consisting of two reactions: auste-
nite-ferrite and austenite-pearlite. Below the stable eutectoid temperature, the ferrite
begins to form. If the transformation has not been completed before the metastableeutectoid temperature is attained, pearlite starts to form and grows competitively
with ferrite. Therefore, the following relation is assumed to hold:
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1627
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
6/36
fl f fg f ff fp 1 S
where fl, f, fg, f, ff and fp are the liquid, austenite, graphite eutectic, cementite
eutectic, ferrite and pearlite volumetric fractions, respectively. For the liquidsolidphase-change ff fp 0
, Eq. (5) reads:
fl f fg f 1 T
whereas for the solidsolid phase change (fl 0, fg cte, i.e. neglecting secondaryand eutectoid graphite precipitations and f cte) it becomes:
f ff fp 1 fg f U
such that only the variable fractions for the respective phase-changes are written on
the left-hand side of these two last expressions. In this context, the full description of
the material behaviour considering the particular response of these constituents in a
unied formulation using standard concepts of the mixing theory is the main feature
of the present constitutive model. Thus, any mixed variable jmx of this compositematerial can be dened by weighting the dierent contributions of each constituent
or component as:
jmx fljlfjfgjgfjffjffpjp V
As usual, in this mixing approach jmx and, therefore, dierent densities canbe considered for each constituent. The density at the initial conguration is also
dened by Eq. (8) but taking the initial volumetric fractions in its calculation.
Moreover, the computation of jcp cp l; ; ; f;p could be carried out by con-sidering additional mass balances between the dierent constituents. Note, however,
that in the constitutive equations described below there is no need to compute jcp asthey may only depend on which is, in fact, obtained through Eq. (1).
It should be mentioned that this approach is a generalization of that dened in
mixed form in terms of the following macroscopic phases (Celentano, 1997): 1)
liquid, 2) mushy zone (liquid+austenite+graphite+cementite), 3) solid (austeni-
te+graphite+cementite), 4) (austenite+graphite+cementite+ferrite+pearlite) and
5) solid (graphite+cementite+ferrite+pearlite).
In what follows any phase-change variable denoted as pc will be decomposed as:
pc pcjpcjgpcjpcjfpcjp W
where the ``a, g, c'' and ``f, p'' terms are respectively related to the liquidsolid and
solidsolid phase-changes.
1628 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
7/36
3.1. Viscoplastic internal variables
A possible choice for the viscoplastic internal variables is: 1
evp, 2
#vp,
3 vp, 4 vp nvpnt 4 with q1 , q2 Cvp, q3 Kvp, q4 T where evp is theAlmansi viscoplastic strain tensor, #vp is the viscoplastic isotropic hardening vari-
able, vp is the viscoplastic kinematic hardening tensor, vp is the viscoplastic yield
entropy, J is the Kirchho stress tensor, Cvp is the viscoplastic isotropic hard-ening function and Kvp is the back stress tensor. This choice is an adaptation to the
present viscoplastic context of that considered by Celentano et al. (1999) for elasto-
plasticity. The corresponding evolution equations, assuming k 1 andgk @=@qk, are dened as:
Lv e
vp
@
@ IH:I
#:
vp @@Cvp
IH:P
Lv vp @
@KvpIH:Q
&: vp @
@TIH:R
where the symbol Lv denotes the well-known Lie derivative (Marsden and Hughes,
1983). Note that Eq. (10.1) denes in fact the evolution of the viscoplastic rate-of-
deformation tensor dvp Lv evp . The evolution equation for vp originally proposedwithin a thermoplasticity framework as consistent with the principle of maximum
plastic dissipation (Armero and Simo, 1993) and extensively used in solidication
problems (Celentano et al., 1996, 1999), is also adopted here.
The viscoplastic potential is assumed to be given by (Hamata, 1992; Celentano,
1997):
tpjmxpc II
where tpjmx and pc are the thermoviscoplastic and phase-change parts of ,respectively. The rst term accounts for classical viscoplastic phenomena and is
written in mixed form through Eq. (8). The phase-change term describes the solid
solid phase transformation viscoplasticity eect during the ferrite and pearlite
transformations occuring in the macroscopic phase 4).
For each component, tpjcp cp l; ; g; ; f;p is written as:
tpjcpFjcp 1nvjcp
1 nvjcp
Kvjcp nvjcp IP
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1629
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
8/36
such that Kvjcp Kvjcp T is the viscosity, nvjcp nvjcp T is a material parameter, hidenotes de MacAuley symbol and Fjcp is the von Mises temperature-dependent yieldfunction dened as:
Fjcp
3J2kp
Cjcp IQ
where J2K 1=2 Kvp HX Kvp H is the second invariant of the deviatoric tensorH Kvp H and Cjcp Cjcp Cvp;T is the total hardening function given by:
Cjcp CthjcpCvp IR
with Cthjcp Cthjcp T being thermal hardening function which, in absence of visco-plastic eects, establishes the elastic domain for a given temperature. It should be
noted that, in general, Cthjcp could be dierent for each component.As stated above, the phase change term pc is dened according to Eq. (9) where a
simpler model, originally validated with experiments by Hamata (1992) for the aus-
tenite-ferrite transformation, is considered choosing cpjph 0 ph ; g; while thesolid-solid phase-change contribution pcjph ph f;p takes the form:
pcjph
J2Kp 1ntjph
1 ntjph
Ktjph
nt f:ph IS
where, once more, the viscosity Ktjph and the material parameter ntjph characterizethe coupled solid viscous behaviour during the ferrite and pearlite transformations.
Note that Eq. (15), which leads to a non associate constitutive model, states that this
phase transformation is considered as a perfect viscoplastic eect that only takes
place when fph ph f;p evolves and, moreover, it aects the evolution laws for evpand vp given by Eqs. (10.1) and (10.3), respectively. Further, the evolution of evp
due to the solid-solid phase-change is, essentially, very similar to dierent simple
transformation-induced plasticity models reviewed by Fischer et al. (1996), i.e. such
evolution is proportional to the deviatoric stress plus the rate of the volumetric
fraction.
3.2. Phase-change internal variables
According to the above considerations, the phase-change internal variables are:
1 f, 2 fg, 3 f, 4 ff, 5 fp npcnt 5
with q1 q, q2 qg, q3 q,q4 qf, q5 qp. As mentioned above, the austenite, graphite and cementite frac-tions are associated to the liquid phase-change while the ferrite and pearlite evolve
during the solid-solid transformation.
In the liquid-solid phase-change, experimental observations suggest that both the
dendritic and eutectic grains are equiaxed except over a small region located near themould walls where columnar grains appear (Chang et al., 1991). In this work all
grains formed are assumed to be equiaxed and spherical in shape for the eutectics or
1630 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
9/36
enveloped within a sphere for the dendritic grains. Although dierent microstructure
models for S.G. cast iron have been proposed taking into account particular kinetic
aspects occuring during the solidication and cooling process (Banerjee and Stefa-
nescu, 1991; Chang et al., 1991), a simpler microstructure model to describe both theliquidsolid and solidsolid phase-changes is considered in this work. This model
includes primary austenite dendrites formation, nucleation and growth laws for the
graphite eutectic, cementite eutectic, ferrite and pearlite fractions.
Considering that during the austenite solidication the carbon diuses very
rapidly in iron, the austenite volumetric fraction can be directly determined by the
inverse lever rule as (Goettsch and Dantzig, 1994):
f 1
1
kH
T TlT
T0
IT
subject to the condition f:50, where k
H kH (%Si) is the equilibrium distributioncoecient which is the relationship between the slope of the liquidus and solidus
curves of the Fe-C diagram, Tl Tl (%C, %Si) is the liquidus temperature, T0 T0 (%C, %Si) is the temperature corresponding to the intersection of the liquidus
and solidus temperature curves and %C and %Si denote the carbon and silicon
contents, respectively. It is important to mention that the evolution of f directly
derived from Eq. (16) does not take into account neither the inuence of the cooling
rate nor the remelting eect (Celentano and Cruchaga, 1999).
The eutectic fractions fg and f can be determined through a nucleation andgrowth laws. Assuming a quasi-instantaneous nucleation and a subsequent spherical
growth, the expression for such fractions is (Goettsch and Dantzig, 1994):
fph 4
3NjphR3jph IU
where Njph is the grain density and Rjph is the grain size ph g; . These variablesare respectively obtained with the nucleation and growth models briey presented
below. For simplicity, the grain impingement eect is not included in Eq. (17) (for
more details, see Chang et al., 1991).The following nucleation law is adopted in the present work:
N: jph Ajphnjph TjphT
njph1 T:h i IVwhere Ajph and njph ph g; are nucleation parameters which depend on %C, %Siand the inoculant content. In Eq. (18), Tg and T denote the graphite (stable) and
cementite (metastable) eutectic temperatures, respectively, which also depend on
%C and %Si. It is well-known that the major eect of silicon is to widen the range
Tg T (Flemings, 1974). As this range increases, the probability of forming gra-phite eutectic rather than a cementite eutectic is also increased. Hence, %Si pro-
motes the formation of graphite. Moreover, Eq. (18) clearly states that the grain
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1631
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
10/36
density is constrained to increase and this fact is only possible for positive under-
coolings (indicated by TjphT
) and negative cooling rates. Between Tg and T only
the graphite eutectic may nucleate while below T the cementite eutectic may also
nucleate. As can be seen, distinct nucleation rates are obtained for graphite andcementite eutectics due to the dierences in the respective undercoolings.
The grain size evolution equation is dened as:
R: jph Bjph TjphT
mjph IWwhere Bjph and mjph (ph g; ) are growth parameters. Once again, note that in thetemperature range Tg T only the graphite eutectic evolution takes place and,therefore, T can be considered as the transition temperature for the solidication of
gray to white iron. Further, Bjg
1=10B
j is chosen in order to represent the
experimental fact showing that the growth rate of graphite is about 1=10 of that of
cementite for equal respective undercoolings (Flemings, 1974). According to Eq. (6),
the liquid-solid phase-change ends once fl 0, i.e. f fg f 1.In the solid-solid phase-change, the modelling of ferrite and pearlite can be also
described by a law similar to Eq. (17) where, in this case, ph f;p. For the ferritetransformation, in particular, the assumptions leading to this expression can be
found in Chang et al. (1991). It should be noted that for low temperature rates
(approximately less than 0.2C) a simplied model of ferrite, based on the AvramiJohnsonMehl equation, can be considered (Hamata, 1992). Moreover, an Avrami
equation can be also used to compute the fraction of pearlite (Chang et al., 1991).However, for simplicity, the unied expression given by Eq. (17) is retained.
The nucleation and growth laws for the ferrite and pearlite transformation are
assumed to be respectively given by expressions similar to Eqs. (18) and (19) written,
for these cases, in terms of the ferrite and pearlite undercoolings and using particular
nucleation and growth parameters ph f;p . These undercoolings are dened withthe ferrite (stable) eutectoid temperature Tf and the pearlite (metastable) eutectoid
temperature Tp. It should be mentioned that other nucleation and growth laws have
been proposed in the literature where the more relevant are a diusion-controlled
growth-rate equation for the ferrite (Chang et al., 1991) and exponential laws based
on the eutectoid undercooling for the nucleation and growth of pearlite (Goettschand Dantzig, 1994). Finally, the solidsolid phase-change nishes, as stated by Eq.
(7), when f 0, that is ff fp 1 fg f.
3.3. Specic free energy function
The following specic free energy function is proposed:
tejmxtpjmxpc PH
where tejmx; tpjmx and pc are the thermoelastic, thermoviscoplastic and phase-change parts of. This partially coupled additive form of is assumed to describe
the isotropic material behaviour in all the above mentioned macroscopic phases
1632 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
11/36
present in the solidication process. In all the equations to be described below, the
subscript 0 indicates the value of the variables at the initial conguration and the
superscript s refers to secant thermomechanical properties measured at the spatial
conguration with respect to the reference temperature Tref. Additionally, e0 0 isconsidered.
The thermoelastic part tejcp cp l; ; g; ; f; p of each component is:
tejcp 1
20e evp X CsjcpX e evp
1
0e evp X CsjcpX ethjcp
cjcpc0 jcp1
0e evp X 0 0 T T0 0jcp PI
where Csjcp Csjcp T is the secant elastic isotropic constitutive tensor and ethjcp isthe Almansi thermal strain tensor given by:
ethjcp1
21 1 athjcp
2=3 1 PP
where athjcp sthjcp T Tref sth0jcp T0 Tref with sthjcp sthjcp T being thesecant volumetric thermal dilatation coecient. Note that if athjcp( 1, ethjcpathjcp=31 which is, in fact, the classical expression for the innitesimal thermal straintensor. The term cjcp is:
cjcp T
Tref
Acjcpd PQ
where Acjcp is a function that can be dened in terms of the secant specic heatcapacity csjcp csjcp T (Celentano et al., 1996).
As usual, the secant elastic isotropic constitutive tensor can be decomposed into
its deviatoric and volumetric parts (Malvern, 1969). According to Celentano et al.
(1996), a particular denition of Cs
jl is adopted here which consists of neglecting itsdeviatoric contribution assuming, therefore, a purely elastic volumetric behaviour
for this phase.
The thermoviscoplastic part of is cp l; ; g; ; f;p :
tpjcp 1
20hCjcp #vp 2
1
20hKjcpvp X vp
1
0T&vp PR
where hC
jcp
hC
jcp T
and hK
jcp
hK
jcp T
are the viscoplastic isotropic and kine-
matic hardening moduli, respectively.The phase-change part of is given by Eq. (9) assuming similar expressions for
the liquid-solid and solidsolid phase-changes, that is ph ; g; ; f;p :
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1633
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
12/36
pcjph Ljphjph PS
with
Ljph Lsjphfph PT
jph 1
0e evp X CsjphX epcjph PU
where Lsjph Lsjph T is the secant specic latent heat and epcjph is the Almansiphase-change strain tensor both associated to the volumetric fraction fph. This
phase-change strain tensor is similarly dened as the Almansi thermal strain tensor
[Eq. (22)] in the form:
epcjph1
21 1 apcjph
2=3 1 athjph 2=3
1 PV
where apcjph spcjphfph and spcjph spcjph T is the secant phase-change volumetricdeformation. As it can be seen, Eqs. (26) and (27) account for latent heat release and
phase-change volumetric deformation, respectively (see Flemings, 1974, for a further
discussion on the physical aspects of these phenomena).The denition of the dierent contributions ofgiven by Eqs. (21), (24) and (27)
consider the density at the initial conguration 0 instead of its current value . This
simplication is consistent with the DoyleEricksen approach (1956) commented in
Section 2. In the constitutive equations, however, any density change is given by J.
Moreover, as mentioned above, the present denition of is an extension, con-
sidering large strains, microstructural phase-changes and mixed response, of the
specic free energy function proposed and used by Celentano et al. (1999).
It should be noted that this large strain context allows to predict the external
(surface) shrinkage occuring in some casting situations. A simple internal shrinkage
prediction criterion due to liquid contraction (Celentano, 1998a) could be includedin Eq. (27) of the present formulation but, for simplicity, is not considered in this
work.
The proposed denition ofallows the derivation of all the constitutive equations
and the internal dissipation described in Section 2. Some details are discussed below.
3.4. Constitutive laws
With the considerations given above, the Cauchy stress tensor is given by:
tejmxpc 0 PW
where tejcp cp l; ; g; ; f;p is:
1634 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
13/36
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
14/36
1990; Baldoni and Rajagopal, 1997) consisting of adding the term 2d ( being the
dynamic viscosity of the uid) to the expression given by Eq. (29) for cp l wouldallow to include this eect. In this context, the resulting viscous term 2d X d should
be added to the internal dissipation equation. For simplicity, however, this viscosityeect is not considered in the uid phase throughout this work.
Further, the expressions for the specic entropy function, the tangent specic heat
capacity, the tangent conjugate of the thermal dilatation tensor, the internal heat
source and the conjugate of the internal variables (note that the relations for and T
are identically fullled) can be straightforwardly derived considering the denition
ofgiven above. Such expressions can be found in Box 1
Box 1. Variables of the S.G. cast iron model
Specic entropy function
tejmxtpjmxpc 0
with (cp l; ; g; ; f;p and ph ; g; ; f;p
tejcp
1
20e
evp
X@Csjcp@T
X e
evp
1
0e
evp
X Cs
jcpX eth
Tjcp1
10
e evp X @Csjcp@T
ethjcpAjcp
tpjcp 1
20
@hCjcp@T
#vp 2 120
@hKjcp@T
vp X vp 10
&vp
pcjph @Lsjph@T
fph 1
0e evp X @C
sjph@T
X epcjph1
0e evp X CsjphX epcT jph1
where
ethTjcp1
31 athjcp 1=3
asthjcp@sthjcp@T
T Tref !
and
epcT jph
1
31 apcjph 1=3@spcjphj
@T1 athjph 2=3
fph 1 1 apcjph 2=3
ethTjph
Tangent specic heat capacity
c ctejmxctpjmxcpc(continued on next page)
1636 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
15/36
with cp
l; ; g; ; f;pndph
; g; ; f;p
ctejcp
T
20e evp X @
2Csjcp@T2
X e evp 2T0
e evp X @Csjcp@T
X ethTjcp1 T
0e evp X CsjcpX
@ethTjcp@T
1 T0
e evp X @Csjcp
@T2
X ethjcpcsjcp@csjcp@T
T Tref
ctpjcp T20
@
2
hCjcp@2T
#vp 2 T20
@
2
hkjcp@2T
vp X vp
cpcjph T@2Lsjph@T2
fph T
0e evp X @
2Csjph@T2
X epcjph2T
0e evp X @C
sjph@T
X 1 2T0
e evp X @Csjph@T
X epcT jph1
T
0e evp X Csjph
@epcT jph@T
1
Tangent conjugate of the thermal dilatation tensor
tejmxpc
with cp l; ; g; ; f;pndph ; g; ; f;p
tejcp1
JCsjcpX ethTjcp1
1
J
@Csjcp@T
X e evp ethjcp
pcjph1
J
@Csjph@T
X epcjph1
JCsjphX epcT jph1
Specic internal heat source
rvpnt
1
0T X Lv evp T
@Cvp
@T Cvp
#:
vp T@Kvp
@T Kvp
X Lv
vp !
Box 1 (continued)
(continued on next page)
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1637
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
16/36
3.5. Dissipation inequality
In this context, the viscoplastic internal dissipation is written as:
Dvpnt
X Lv evp Cvp#:
vp Kvp X Lv vp T&: vp QP
and the corresponding phase-change contribution is ph ; g; ; f;p :
Dpcnt
jph 0 Lsjph1
30e evp X CsjphX 1 apcjph
1=3spcjph1
!f:ph QQ
such that, as mentioned before, Dvpnt
D
pcnt50. It is worth noting that the volumetric
fraction derivatives oftejmx and tpjmx have not been considered in Eq. (33) as theyare assumed to be negligible in comparison to the latent heat and phase-change
volumetric conjugate variables of fph indicated in the bracketed terms.
rpcnt
jph
4 T@Lsjph@T
Ls
jph
T
0
e
evp
X@Csjph@T
X epcf
jph1
10
e evp X CsjphX epcf jph1
T0
e evp X CsjphX@e
pcf
jph@T
1
5f:ph forph ; g; ; f;p
and
e
pc
f jph1
3 1 apcjph 1=3
s
pcjph
Conjugate of the internal variables
Cvp Cvptp jmx whereCvptp jcp hjcp#vp for cp l; ; g; ; f;p
Kvp
Kvp
tp jmx whereKvp
tp jcp hujcpvp
for cp l; ; g; ; f;pqph 0Lsjph e evp X CsjphX epcf jph1 forph ; g; ; f;p
Box 1 (continued)
1638 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
17/36
It has been shown (Celentano et al., 1996, 1999) that the condition Dvpnt50 is
satised if two sucient conditions related to the isotropic hardening behaviour and
the thermal softening eect are assumed: (i) h50 and (ii) @Cth
jcp=@T50
cp l; ; g; ; f;p . Knowing that the rst term in Eq. (33) (with Lsjph50ph ; g; ; f;p ) is in general more relevant that the second one in many castingsituations, the fulllment of the condition D
vpnt
Dpcnt50 can be guaranteed.Moreover, it should be noted that the second term of Eq. (33) contains a trans-
formation (phase-change) work since it can be written, with the help of Eq. (30), as:
J=3 tejph1=JCsjphX ethjph
X 1 apcjph 1=3
spcjph1, where the product of mean ther-moelastic stress and phase-change volumetric deformation for each phase can be
seen. A similar expression of this conjugate variable to fph using dierent state vari-
ables in the specic free energy denition has been reported by Levitas (1998) for
martensitic and other transformations.
Finally, a mixed form for the conductivity coecient is assumed, i.e. k kjmx,such that kjcp50 cp l; ; g; ; f;p in order to verify the thermal dissipation men-tioned in Section 2.
4. Green sand model
Green sand, as usually prepared and used in foundries, is a ne porous mixture of
approximately 87 wt.% of silica grains, 7 wt.% of clay, 3 wt.% of carbon black and3 wt.% of free water. Thus,
fm fv 1 QR
where fm and fv respectively denote the matter (silica grains+clay+carbon black)
and pore volumetric fractions. Typical compaction (the compaction of sand during
the manufacturing process of the mould is not studied here) corresponds to an
overall density of 1600 kg/m3, an average value of the porosity fv of 0.26 and a water
saturation of 0.18 (Azzouz, 1995). Considering that the pores are partially lled with
water, the value fw0 0:260:18 0:0468 is therefore taken as an initial conditionfor the free water volumetric fraction fw which approximately leads to a maximum
stress peak in the compression test.
Complex phenomena occur in the sand when it is subjected to thermomechanical
loading imposed during casting (Azzouz, 1995; Ami Saada et al., 1996). As a rst
approach to the problem, however, most of them are considered through the
dependence of the constitutive laws upon temperature and only the vaporization of
free water as an eect related to the disappearing of initial humidity will be included
in the model. Moreover, an eventual condensation of water steam is not considered.
In this green sand constitutive model, the viscoplastic and phase-change eects are
mainly related, respectively, to damage (degradation of the material) and hygro-metric (matrix suction due to the presence of free water) phenomena occuring in the
material during the process.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1639
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
18/36
Once more, the specic free energy function is additively decomposed into a
thermoelastic, thermoplastic and hygrometric contributions. Moreover, a non
associate form of the viscoplastic potential , assumed to be independent on the
hygrometric eect, is considered for the viscoplastic evolution equations (Azzouz,1995; Celentano, 1997). For simplicity, the eect of the free water is neglected in the
denition ofte and tp and it is only considered in pc. As no mixing law is applied
to those terms, the variations of the thermomechanical properties with fw are simply
taken into account by means of temperature-dependent laws.
4.1. Viscoplastic internal variables
The following set of viscoplastic internal variables is adopted: 1 evp, 2 dvp,3
vp n
vp
nt 3 with q1 , q2 Yvp, q3 Twhere the same notation of previousSections is used and, additionally, dvp is the viscoplastic damage variable and Yvp is
the viscoplastic damage function. The corresponding evolution equations, assuming
gk @=@qk are
Lv evp @
@QS:I
d:
vp @@Yvp
QS:P
&: vp @
@TQS:Q
where, as shown below, both isotropic and kinematic hardening eects are not
dened by evolution laws because they can be considered as a function of and dvp.
The function is dened as:
Fh inv
KvQT
such that Kv Kv T is the viscosity, nv nv T is a material parameter and F is theyield function given by:
F f1I21 f2J2 f3I1 C QU
where I1 tr (tr being the trace symbol), J2 1=2H X H is the second invariant ofthe deviatoric tensor H, f1 12 K 1 dvp 2, f2 3=G 1 dvp 2, f3 hu= 1 dvp (K,G and hu being the bulk, shear and kinematic hardening moduli, respectively) and
C C dvp;T is the total hardening function given by:C Cth hdvp QV
1640 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
19/36
with Cth Cth T being the thermal hardening function and hC is the isotropic hard-ening modulus. Note that for small values of Kv, i.e. Kv 3 0, the rate-independentplasticity is directly recovered. This expression of F corresponds to an ellipsoid in
the Westergaard stress-space. Conversely to a classical DruckerPrager criterion,this yield function is able to describe the inelastic behaviour which is induced by the
degradation of the material observed in the case of loadings close to the hydrostatic
pressure. The term linearly dependent on I1 introduces a non-symmetric elastic
domain in tension and compression (Azzouz, 1995).
The viscoplastic potential is:
F hdYvp QW
where hd is a material property associated to the material degradation (Azzouz,
1995).
4.2. Hygrometric internal variable
The vaporization of free water is described by 1 fw npcnt 1
with q1 qw. Theevolution of fw is assumed to depend exclusively on T. Hence,
f:w H fw
fw0Tv T0
T:
h iRH
where His the Heaviside function and Tv is the water vaporization temperature. The
irreversible evolution of fw can clearly be noted in this expression.
4.3. Specic free energy function
As mentioned above, the proposed specic free energy function that governs the
isotropic material behaviour during the process is written as:
te tp pc RIwhere te, tp and pc are the thermoelastic, thermoviscoplastic and hygrometric
parts of.
The thermoelastic part te is:
te 1
20e evp X 1 dvp Cs X e evp 1
0e evp X 1 dvp Cs X eth
c c0 1
0e evp X 0 0 T T0 0 RP
which is, in fact, similar to Eq. (21) for a single-component material taking into
account the degradation factor 1 dvp in the elastic constitutive tensor.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1641
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
20/36
In the absence of viscoplastic hardening eects, the thermoviscoplastic part of is
simply written as:
tp 10
T&vp RQ
The hygrometric part of is given as:
pc L L0 RR
with
L
Lsfw
RS
1
0e evp X 1 dvp Cs X epc RT
where the expressions of the variables involved correspond to those presented above
for a single-component material. As in the previous constitutive model, Eqs. (45)
and (46) include the latent heat release and hygrometric volumetric deformation,
respectively. In this case, note that the initial hygrometric eect is explicitly con-
sidered in Eq. (44) and, therefore, is not included in the initial stress of Eq. (42).
Further, the consideration of 0 in Eqs. (42), (43) and (46) is based on the samearguments stated above for the S.G. cast iron model.
Once more, as shown below, with the proposed denition of it is possible to
obtain all the constitutive equations and the internal dissipation corresponding to
this model.
4.4. Constitutive laws
Making use of the proposed specic free energy function the Cauchy stress tensor
results:
1J
1 dvp Cs X e evp eth epc 0 RUwhere the thermoelastic and hygrometric terms can easily be identied and, further,
the additive decomposition of the Almansi strain tensor is apparent, that is, ee e evp eth epc derived from F Fe Fp Fth Fpc taking into account the samearguments commented for the S.G. cast iron described above. Once again, Eq. (47)
is only valid for small elastic strains and, therefore, this restriction is assumed in the
deformation of the sand during the casting process.
The expressions for the specic entropy function, the tangent specic heat capa-city, the tangent conjugate of the thermal dilatation tensor, the internal heat source
and the conjugate of the internal variables can be found in Box 2.
1642 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
21/36
Box 2. Variables of the green sand model.
Specic entropy function
te tp pc 0
with
te 1
20e evp X 1 dvp @C
s
@TX e evp 1
0e evp X 1 dvp Cs X ethT1
1
0e evp X 1 dvp
@Cs
@T X eth Ac
tp 1
0&vp
pc @Ls
@Tfw 1
0e evp X 1dvp @C
s
@TX epc 1
0eevp X 1 dvp Cs X epcT 1
where
ethT 1
3 1 ath 1=3 sth @s
th@T T Tref
!
and
epcT
1
31 apc 1=3@spc
@T1 ath fw 1 1 apc
2=3 ethT
Tangent specic heat capacity
c cte ctp cpc
with
cte 1
20e evp X 1 dvp @
2Cs
@T2X e evp 2T
0e evp X 1 dvp @C
s
@T
X ethT1 T
0e evp X 1 dvp Cs @e
thT
@T1 T
0e evp X 1 dvp @
2Cs
@T2
X eth
cs
@cs
@T
T
Tref
(continued on next page)
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1643
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
22/36
ctp 0
cpc T@2Ls
@Tfw
T
0e evp X 1 dvp @
2Cs
@T2X epc 2T
0e evp X 1 dvp @C
s
@T
X epcT 1
T
0e evp X 1 dvp Cs X @e
pcT
@T1
Tangent conjugate of the thermal dilatation tensor:
te
pc
with
te 1
J1 dvp Cs X ethT1
1
J1 dvp @C
s
@TX e evp eth
pc 1
J1 dvp @C
s
@TX epc 1
JCs X e
pcT 1
Specic internal heat source:
rvpnt
10
T X Lv evp
rpcnt
T@Ls
@T Ls T
0e evp X 1 dvp @C
s
@TX e
pcf 1
1
0e evp
X 1 dvp Cs X epcf 1 T
0e evp X 1 dvp Cs X @e
pcf
@1
!f:
w
and
epcf
1
31 apc 1=3
spc
Conjugate of the internal variables
Yvp 12
e evp X Cs X e evp e evp X Cs X eth e evp X Cs X epc
qw 0Ls e evp X 1 dvp Cs X epcf 1
Box 2 (continued)
1644 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
23/36
4.5. Dissipation inequality
The derivation of the internal dissipation gives:
Dnt X Lv evp T&: vp Yvpd:
vp
0 Ls 1
30e evp X 1 dvp Cs X 1 apc
1=3spc1
!f:w50 RV
which clearly shows the viscoplastic and hygrometric contributions. Similar con-
siderations to those commented in Section 3.5 are also valid for the fulllment of
this dissipation inequality and the transformation work included in the conjugate
variable to fw.
5. Finite element formulation
This section briey describes the nite element equations derived from the pro-
posed thermomechanical/microstructural formulation presented above and, further,
some specic details of the solution strategy followed to solve the resulting coupled
system of discretized equations.
Assuming standard spatial interpolations for the displacement and temperature
elds and following the typical procedures within the nite element context (Zien-kiewicz and Taylor, 1989), the global discretized thermomechanical equations (also
including microstructural eects) can be written in matrix form for a certain time t
as:
RU FU Ff MU F 0RT FT CT
: KT Lnt GU: 0 RW
where RU and RT are the mechanical and thermal residual vectors, respectively.
Moreover, FU is the external force vector, Ff is the mechanical contact vector, M isthe mass matrix, U is the nodal displacement vector, F denotes the internal force
vector, FT is the external heat ux vector, C is the capacity matrix, T is the nodal
temperature vector, K is the conductivity matrix, Lnt is the internal heat ux vector
(which can be decomposed into Lnt Lvpnt Lpcnt according to the denition of rnt)and G is the thermoelastic coupling matrix. The element expressions of these matri-
ces and vectors are very similar to those derived by Celentano et al. (1996) in an
innitesimal context using dierent material constitutive models. Nevertheless, the
extension of such expressions to the formulation presented in this work can be
straightforwardly performed.
The integration of the terms containing time derivatives ofU and T in system (49)is carried out with the Newmark method and the generalized mid-point rule algo-
rithm, respectively (Zienkiewicz and Taylor, 1989). The latter has been also used to
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1645
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
24/36
integrate all the rate equations involved in the S.G. cast iron and green sand models
presented above. Note that in this phenomenological approach the time integration
of both the viscoplastic and phase-change internal evolution laws is performed at the
same integration points.A detailed description of the mechanical and thermal boundary conditions is also
reported by Celentano et al. (1996). It is important to remark, however, the con-
sideration of a gap and/or pressure dependent heat transfer coecient between the
casting and the mould. This coupling eect can be an extremely important fact in
many casting simulations.
The numerical solution of system (49) is achieved using a staggered scheme con-
sidering an improved isothermal split (Celentano et al., 1999). In this context, dif-
ferent ways to solve the mechanical and thermal problems (given by RU and RT,
respectively) are available. In this work, in particular, the well-known total Lagran-
gean approach under isothermal conditions has been considered for the mechanical
problem while the solution of the thermal problem is done at the spatial congura-
tion assuming xed elastic congurations for each component of the S.G. cast iron
and the sand. This methodology is stable and preserves the coupling degree of the
formulation, i.e., the evaluation of the coupling terms is not shifted in time in order
to obtain xed contributions.
Finally, in order to overcome the volumetric locking eect on the numerical
solution when incompressible viscoplastic ows are considered, an improved strain-
displacement matrix is proposed in this work. Based on the deformation gradient
multiplicative standard decomposition into deviatoric and volumetric parts, andassuming a selective (or averaged or smoothed) numerical integration for the volu-
metric part of F, the derived strain-displacement matrix, directly obtained by line-
arization of the Green-Lagrange strain tensor, can be classied within the so-called
B-bar methods originally developed for innitesimal strains by Hughes (1980). The
expressions of this matrix for the 2D, axisymmetric and 3D cases are given in Box 3.
Box 3. Strain-displacement matrix B" for large strain analysis
For a given node
B" J"
J
2 32=3Bst Bm
where:
J" det F"
with F" FdevF"vol such that:Fdev
J1=3F X deviatoric part of F
F"vol J"1=31: volumetric part of F numerically integrated in a selective(or averaged or smoothed) form (continued on next page)
1646 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
25/36
As recently reported by Celentano (1998b), this proposed approach has been suc-
cessfully checked in the analysis of classical large strain benchmark problems.
Moreover, it can be demonstrated that this matrix recovers the correspondingexpression of the innitesimal case, based on the additive decomposition of the
strain tensor into deviatoric and volumetric parts, when F 3 1.
Bst: standard straindisplacement matrix for large strain analysis (see Bathe,
1981)
Bm: improved contribution
Bm 1
3
C11AU2D C11AV2DC22AU2D C22AV2D
2C12AU2D 2C12AV2D
PR
QS 2se
Bm 1
3
C11AUax C11AVaxC22AUax C22AVax
2C12AUax 2C12AVax
C33AUax C33AVax
P
TTR
Q
UUSxsymmetrse
Bm 1
3
C11AU C11AV C11AWC22AU C22AV C22AW
2C12AU 2C12AV 2C12AWC33AU C33AV C33AW
2C13AU 2C13AV 2C13AW2C23AU 2C23AV 2C23AW
PTTTTTTR
QUUUUUUS
3se
with
C FTF: right CauchyGreen deformation tensor (: transpose symbol)
A 1
J"
@J"
@ 1
J
@J
@ U;V;W
U;V;W: components of the nodal displacement vector U
A2D AjF13F23F31F320YF331 U;VAax Aj
F13F23F31F320YF331NU
X
U;V(U: radial nodal displacement)
N: shape function for displacements
X: radial material coordinate
Box 3 (continued)
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1647
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
26/36
6. Analysis of a solidication test
The analysis of a cylindrical casting specimen of S.G. cast iron (diameter=70 mm
and height=140 mm) in a green sand mould surrounded by a steel shell (internaldiameter=185 mm, thickness=30 mm and height=260 mm) is performed. This
problem has been extensively studied using simplied innitesimal strains con-
stitutive models for the materials involved (Celentano et al., 1995; Celentano, 1997).
The experimental apparatus is schematically shown in Fig. 1. Both temperature and
radial displacement evolutions have been measured during solidication and cooling
approximately at the midheight of the specimen (BRITE/EURAM Synthesis
Report, 1994). Thermocouples were placed on three radial directions at 0, 120 and
240, starting from the cylinder central axis to the surrounding sand mould in orderto visualize the thermal gradient evolution. Radial displacement were measured at
the same directions on the cylinder external skin using silica rods.
Fig. 1. Solidication test: experimental apparatus.
1648 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
27/36
Table 1
Material propertiesa of S.G. cast iron
Young's modulus (MPa):
10 at 1150C (1) 44,260 at 750C (a) 500 (g) 163,471 at 20C (c, f, p)10 at 1400C 43,790 at 770C 163,113 at 100C
44,360 at 800C 160,174 at 200C45,935 at 830C 151,650 at 300C42,935 at 850C 135,276 at 400C35,435 at 900C 110,898 at 500C28,435 at 1000C 81,386 at 600 C
5000 at 1100C 52,021 at 700C46,668 at 720 C
Poisson ratio: 0.33 (1, a, g, c, f, p)
Secant thermal dilatation coecient (1/C):20.0106 at 1155C (1) 13.0106 at 20C (a, g, c, f, p)20.0106 at 1400C 12.0106 at 750C
12.0106 at 800C17.0106 at 1000C
Thermal hardening function (MPa):
0.000 1 (1), 260 at 20C (a, g, c, f, p)250 at 200C210 at 400C
60 at 600C50 at 700C30 at 900C20 at 1000C
2 at 1141C0.1 at 1155C
Viscoplastic parameters:
Viscosity (MPa s): 100.0 (1, a, g, c, f, p)
Exponent: 1.0 (1, a, g, c, f, p)
Solidsolid phase-change parameters:
Viscosity (MPa s): 0.0 (f, p)
Exponent: 1.0 (f, p)
Isotropic hardening modulus (MPa): 0.0 (l, a, g, c), 300 (f, p)
Kinematic hardening modulus (MPa): 0.0 (l, a, g, c, f, p)
Secant phase-change volumetric deformation: 0.01 (a, g, c), 0.005 (f, p)
Density (initial) (kg/m3): 6700 (1)
Secant specic heat (J/kg C):917 (1) 740 at 732C (a) 705 (g, c) 540 at 25C (f, p)
705 at 1136C 732 at 723C917 at 1145C
Conductivity (J/m s C):33 at 1250C (1) 35 at 750C 30 (g, c) 27 at 20C (f, p)
33 at 1300C 33 at 1140C 30 at 328C92 at 1350C 30 at 500C
(continued on next page)
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1649
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
28/36
The material thermomechanical properties for the S.G. cast iron and green sand
can be found in BRITE/EURAM Synthesis Report (1994), Hamata (1992) and
Azzouz (1995). The constants involved in the microstructural model for the S.G.
cast iron are equivalent to those used by Chang et al. (1991), Hamata (1992),
Goettsch and Dantzig (1994) and Celentano and Cruchaga (1999). According toSchmitt at al. (1997), the elastic mechanical properties (Young's modulus and Pois-
son ratio) of cementite, ferrite and pearlite have been assumed to be equal in the
present analysis. For the steel, the same S.G. cast iron model was used considering a
single-component material without, for simplicity, hardening and phase-change
eects. All the material properties considered in the simulation are shown in
Tables 14.
The axisymmetric numerical computation used 540 four-noded isoparametric ele-
ments and a time step of 50s. The analysis starts with the mould cavity completely
lled with molten metal at rest at 1250C (i.e. instantaneous lling is assumed) and
22C for the sand and steel moulds. The mould is simply supported at the bottomand convectionradiation conditions have been considered between the external face
of the mould and the environment. The boundary conditions and the nite element
mesh used are plotted in Fig. 2.
The experimental temperature evolution at dierent radial positions are plotted in
Fig. 3. The numerical results obtained with the proposed formulation are also
included for comparison. A good overall agreement can be observed where, more
specically, the liquid-solid and solidsolid phase-changes in the casting are reason-
ably well described.
Moreover, the experimental and numerical radial displacement evolutions at the
casting-mould interface are shown in Fig. 4. The dierent expansion/contractionbehaviours related to the phase-changes occuring during the process can clearly be
seen: (a) contraction till the beginning of the solidication, (b) expansion during
Table 1 (continued)
186 at 1375C 25 at 600C280 at 1400C 32 at 700C
Secant latent heat (J/kg): 263,000 (a), 233,000 (g), 213,000 (c), 16,300 (f) 85,800 (p)
Composition: 3.5%C, 2.0%Si, 0.0% inoculant
Equilibrium temperatures (C): 1588 (0), 1179 (1), 1155 (g), 1056 (c), 768 (f), 740 (p)
Partition coecient: 0.46
Nucleation constants:
Parameter (nuclei/m3 C3.61): 7.12103 (g, c, f, p)Exponent: 3.61 (g, c, f, p)
Growth constants:
Parameter (m/s C2): 1.6106 (g,f), 1.6105 (c,p)Exponent: 2.0 (g, c, f, p)
a The variations of these properties with the temperature have been assumed to be piecewise linear
within the mentioned temperatures. Below the lowest and above the highest temperatures, the properties
are assumed to remain constant at the same value dened for the extreme temperature.
1650 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
29/36
solidication (graphite precipitation), (c) contraction from the end of the solidica-
tion up to the beginning of the eutectoid transformation, (d) second expansion dur-
ing the eutectoid transformation and (e) nal contraction to room temperature.
Almost identical behaviours have been observed for the three directions mentioned
above. Some dispersion in the experimental results was detected, however, and an
average curve has been included in Fig. 4. The diculties in the measurement tasks
may be presumably attributed to dierential vertical dilatation between the castingand sand mould (see Fig. 5) leading to a unacceptable shear force in the silica rods.
Although the numerical tting is only qualitative, the response provided by the S.G.
Table 2
Material propertiesa of green sand
Young's modulus (MPa): 150
Poisson ratio: 0.33
Secant thermal dilatation coecient (1/C): 13.0106
Thermal hardening function (MPa): 5.0
Viscoplastic parameters:
Viscosity (MPa s): 0.0
Exponent: 1.0
Isotropic hardening modulus (MPa): 0.0
Secant phase-change volumetric deformation: 0.01
Density (initial) (kg/m3): 1550
Secant specic heat (J/kg C):917 at 20C917 at 200C1025 at 400C1063 at 700C1100 at 1000C1100 at 1300C
Conductivity (J/m s C):0.86 at 20C0.72 at 200C0.52 at 400C0.61 at 700C0.78 at 1000C0.78 at 1300C
Secant latent heat (J/kg): 3750
Vaporization temperature (C): 100
Initial free water volumetric fraction: 0.0468
a The variations of these properties with the temperature have been assumed to be piecewise linear
within the mentioned temperatures. Below the lowest and above the highest temperatures, the properties
are assumed to remain constant at the same value dened for the extreme temperature.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1651
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
30/36
model proposed in this work correctly reproduces the distinct behaviours observed
at dierent stages of the process. Further, the dierence between the displacement
curves corresponding to the casting and sand gives the normal gap evolution which,
as mentioned in Section 5, usually aects the heat transfer conditions at the casting-
sand interface. This last eect can be also seen in Fig. 5 which depicts the deformed
congurations at four times of the analysis.
The volumetric fraction evolutions of the casting for two radial distances at the
midheight of the specimen are plotted in Figs. 6 and 7. The liquidsolid and solid
solid phase-changes are apparent. It should be mentioned that a low cooling rate has
been obtained promoting the formation of graphite at the expense of cementite and,moreover, completely inhibiting the development of pearlite since all the austenite is
transformed into ferrite.
Table 3
Material properties of steel
Young`s modulus (MPa): 210000
Poisson ratio: 0.3
Secant thermal dilatation coecient (1/C): 12.0106
Thermal hardening function (MPa): 210
Isotropic hardening modulus (MPa): 0.0
Kinematic hardening modulus (MPa): 0.0
Density (initial) (kg/m3): 7900
Secant specic heat (J/kg C): 490
Conductivity (J/m s C): 35
Table 4
Properties at the interfacesa
Heat transfer coecient at the casting-sand interface (J/m2 s C):1000 at gn=0.000 m (gn=normal gap)
100 at gn=0.001 m
Heat transfer coecient at the sandsteel interface (J/m2 s
C): 2000
Heat transfer coecient at the steelair interface (J/m2 s C): 20
Environmental temperature (C): 20
Normal asperity at the castingsand and sandsteel interfaces (MPa/m): 10.010 (high value to avoid
penetration between bodies, see Celentano et al., 1996)
a The variation of the heat transfer coecient with the normal gap has been assumed to be piecewise
linear within the mentioned normal gaps. Below the lowest and above the highest normal gaps, the
properties are assumed to remain constant at the same value dened for the extreme normal gap.
1652 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
31/36
Fig. 2. Solidication test: (a) boundary conditions and (b) nite element mesh.
Fig. 3. Solidication test: temperature evolutions for dierent radial positions at the midheight of the
specimen.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1653
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
32/36
Finally, Fig. 8 shows the free water volumetric fraction evolution for two radial
positions in the sand mould at the same height mentioned above. As expected, the
vaporization starts in regions near the casting-mould interface and progressively
advances towards the outer surface.
Fig. 4. Solidication test: radical displacement evolutions at the midheight of the castingsand interface.
Fig. 5. Solidication test: deforned congurations at times (a) 200 s; (b) 500 s; (c) 1000 s and (d) 3500 s
(amplication factor=10).
1654 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
33/36
Fig. 6. Solidication test: volumetric fractions in the casting for radius=0 mm at the midheight of the
specimen.
Fig. 7. Solidication test: volumetric fractions in the casting for radius=30 mm at the midheight of the
specimen.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1655
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
34/36
7. Conclusions
A large strain thermoviscoplastic formulation for the analysis of the solidication
process of S.G. cast iron in green sand moulds have been presented. This formula-
tion accounts for thermomechanical as well as microstructural behaviours of these
materials in an unied framework allowing, therefore, to analyze the dierent
coupled phenomena occuring in complex casting problems. Several original aspects
of such formulation have been discussed. For the S.G. cast iron model, in parti-
cular, the incorporation of latent heat release and volumetric expansions due to all
metallurgical transformations existing during the solidication and cooling pro-cesses are relevant features of the presented formulation. On the other hand, the
consideration of the hygrometric and damage phenomena stands out in the green
sand model.
The corresponding nite element model has been also derived and briey pre-
sented. Some strategies to achieve the numerical solution have been also proposed.
This formulation has been used in the analysis of a solidication test of S.G. cast
iron in a green sand mould. The model has been partially validated with some
available experimental measurements where reasonable agreement between numer-
ical and experimental results can be observed. However, the diculties associated to
the full material characterization lead to a further research in the thermomechanical/microstructural simulation of solidication processes with the sake of constituting a
robust tool for casting design.
Fig. 8. Solidication test: free water volumetric fraction in the sand for r=45 mm and r=75 mm at the
midheight of the specimen.
1656 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
35/36
Acknowledgements
The support provided by CONICYT (FONDECYT Project No. 1990588) and
DICYT-USACH are gratefully acknowledged.
References
Agelet de Saracibar, C., Cervera, M., Chiumenti, M., 1999. On the formulation of coupled thermoplastic
problems with phase-change. Int. J. Plasticity 15, 1.
Ami Saada, R., Bonnet, G., Bouvard, D., 1996. Thermomechanical behavior of casting sands: experi-
ments and elastoplastic modeling. Int. J. Plasticity 12, 273.
Armero, F., Simo, J., 1993. A priori stability estimates and unconditionally stable product formula algo-
rithms for non-linear coupled thermoplasticity. Int. J. Plasticity 9, 149.
Arnold, S., Saleeb, A., 1994. On the thermodynamic framework of generalized coupled therrnoelastic-viscoplastic-damage modeling. Int. J. Plasticity 10, 263.
Azzouz, F., 1995. Modelisation du comportement thermo-hydro-mecanique d'un sable de fonderie, The se
de Doctorat, Universite Paris 6, France.
Baldoni, F., Rajagopal, K., 1997. A continuum theory for the thermomechanics of solidication. Int. J.
Non-Linear Mechanics 32, 3.
Bammann, D., Chiesa, M., Johnson, G., 1995. A state variable damage model for temperature and strain
rate dependent metals. In: Rajendran, A., Batra, R. (Eds.), Constitutive Laws: Theory, Experiments
and Numerical Implementation. CIMNE, pp. 8495.
Banerjee, D., Stefanescu, D., 1991. Structural transitions and solidication kinetics of SG cast iron during
directional solidication experiments. AFS Transactions 104, 747.
Bathe, K., 1981. Finite Element Procedures in Engineering Analysis. Prentice-Hall.
Bellet, M., Decultieux, F., Me nal, M., Bay, F., Levaillant, C., Chenot, J., Schmidt, P., Svensson, I., 1996.Thermomechanics of the cooling stage in casting processes: three-dimensional nite element analysis
and experimental validation. Metall. Trans. B 27B, 81.
BRITE/EURAM Synthesis Report, 1994. Development of a user-oriented CAB system for simulating the
forming of ductile iron parts (Project No. BE-4596).
Casey, J., 1998. On elastic-thermo-plastic materials at nite deformations. Int. J. Plasticity 14, 173.
Cassenti, B., Annigeri, B., 1989. Thermodynamic constraints on stress rate formulations in constitutive
models. Comput. Mech. 4, 429.
Celentano, D., 1997. A thermomechanical formulation for the solidication process of S .G. cast iron in a
green sand mould. Proceedings of COMPLAS V, p. 1284.
Celentano, D., 1998a. Shrinkage prediction in the thermomechanical analysis of castings. In: Proceedings
of the VIIIth Engineering Foundation Conference on Modeling of Casting, Welding and Advanced
Solidication Processes, The Minerals, Metals and Materials Society, London.
Celentano, D., 1998b. VULCAN: coupled thermomechanical nite element analysis for solidication
problems. User's and Verication Manuals (version 2.3).
Celentano, D., Cruchaga, M., 1999. A thermally coupled ow formulation with microstructural evolution
for hypoeutectic cast-iron solidication. Metall. Trans. B 30B, 731.
Celentano, D., Gunasegaram, D., Nguyen, T., 1999. A thermomechanical model for the analysis of light
alloy solidication in a composite mould. Int. J. Solids Struct. 36, 2341.
Celentano, D., Oller, S., On ate, E., 1996. A coupled thermomechanical model for the solidication of cast
metals. Int. J. Solids Struct. 33, 647.
Celentano, D., Visconte, D., Dardati, P., Oller, S., On ate, E., 1995. A thermomechanical model for
solidication problems: experimental validation. Proceedings of COMPLAS IV, p. 2385.
Coleman, B., Gurtin, M., 1967. Thermodynamics with internal state variables. The Journal of Chemical
Physics 47, No. 2.
Chang, S., Shangguan, D., Stefanescu, D., 1991. Prediction of microstructural evolution in SG cast iron
from solidication to room temperature. AFS Transactions 99, 531.
D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658 1657
8/9/2019 A Large Strain Thermoviscoplastic Formulation for the Solidification Os Sg Cast Iron in a Green Sand Mould
36/36
Cherkaoui, M., Berveiller, M., Sabar, H., 1998. Micromechanical modeling of martensitic transformation
induced plasticity (TRIP) in austenite single crystals. Int. J. Plasticity 14, 597.
Doyle, T., Ericksen, J., 1956. Nonlinear elasticity. Advances in Applied Mechanics 4.
Fischer, F., Sun, Q., Tanaka, K., 1996. Transformation-induced plasticity (TRIP). Appl. Mech. Rev. 49,317.
Flemings, M., 1974. Solidication Processing. McGraw-Hill, New York.
Garcia Garino, C., Oliver, J., 1992. A numerical model for elastoplastic large strain problems funda-
mentals and applications. Proceedings of COMPLAS II, p. 117.
Garcia Garino, C., 1993. A numerical model for the analysis of elastoplastic solids subjected to large
deformations. PhD thesis (in Spanish), Universidad Polite cnica de Catalun a, Spain.
Goettsch, D., Dantzig, J., 1994. Modeling microstructure development in gray cast irons. Metall. Trans.
A 25A, 1063.
Green, A., Naghdi, P., 1971. Some remarks on elastic-plastic deformation at nite strains. International
Journal of Engineering Sciences 9, 1219.
Hamata, N., 1992. Modelisation du couplage entre l'elasto-viscoplasticite anisotherme et la transfor-
mation de phase d'une fonte G.S. ferritique. The se de Doctorat, Universite Paris 6, France.Hughes, T., 1980. Generalization of selective integration procedures to anisotropic and nonlinear media.
Short Communications, 1413.
Inoue, T., Ju, D., 1992. Simulation of solidication and viscoplastic stresses during vertical semicontinuous
direct chill casting of aluminium alloy. Int. J. Plasticity 8, 161.
Kleiber, M., 1975. Kinematics of deformation processes in material subjected to nite elastic-plastic
strains. International Journal of Engineering Sciences 13, 155.
Levitas, V., 1996. Large Deformation of Materials with Complex Rheological Properties at Normal and
High Pressure. Nova Science, New York.
Levitas, V., 1998. Thermomechanical theory of martensitic phase transformation in inelastic materials.
Int. J. Solids Struct. 35, 889.
Levitas, V., Idesmas, A., Stein, E., 1998. Finite element formulation of martensitic phase transitions in
elastoplastic materials. Int. J. Solids Struct. 35, 855.Lubliner, J., 1985. Thermomechanics of Deformable Bodies. Department of Civil Engineering, University
of California, Berkeley.
Lubliner, J., 1990. Plasticity Theory. Macmillan, New York.
Malvern, L., 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood
Clis.
Marsden, J., Hughes, T., 1983. Mathematical Foundations of Elasticity. Prentice Hall, Englewood Clis.
Murnagaham, F., 1937. Finite deformations of an elastic solid. Am. J. Math. 59, 235.
Onsien, M., Grong, ., Gundersen, ., Skaland, T., 1999. A process model for the microstructure
evolution in ductile iron: Part I. The model. Metall. Trans. A 30A, 1053.
Perzina, P., 1971. Thermodynamic theory of viscoplasticity. Advances Appl. Mech. 9, 243.
Schmitt, C., Lipinski, P., Berveiller, M., 1997. Micromechanical modelling of the elastoplastic behavior of
polycrystals containing precipitates-application to hypo- and hyper-eutectoid steels. Int. J. Plasticity 13,
183.
Simo, J., Ortiz, M., 1985. A unied approach to nite deformation elastoplastic analysis based on the use
of hyperelastic constitutive equations. Computer Methods in Applied Mechanics and Engineering 49,
221.
Trovant, M., Argyropoulos, S., 1996. Mathematical modeling and experimental measurements of
shrinkage in the casting of metals. Canadian Metallurgical Quaterly 35, 75.
Wriggers, P., Miehe, C., Kleiber, M., Simo, J., 1989. On the coupled thermomechanical treatment of
necking problems via nite-element-methods. Proceedings of COMPLAS II, p. 527.
Zabaras, R., Ruan, Y., Richmod, O., 1990. Front tracking thermomechanical for hypoelastic-viscoplastic
behavior in a solidifying body. Computer Methods in Applied Mechanics and Engineering 81, 333.
Zienkiewicz, O., Taylor, R., 1989. The nite element method, 4th Edition. McGraw-Hill, London (Vols. 1and 2).
1658 D.J. Celentano / International Journal of Plasticity 17 (2001) 16231658