On the constitutive modeling of coupled thermomechanical phase-change problems

Transcription

1 International Journal of Plasticity 17 21) 1565± On the constitutive modeling of coupled thermomechanical phase-change problems C. Agelet de Saracibar *, M. Cervera, M. Chiumenti ETS Ingenieros de Caminos, Canales y Puertos, International Center for Numerical Methods in Engineering, CIMNE, Edi io Cl, Campus Norte, UPC, Jordi Girona 1-3, 834 Barcelona, Spain Received in nal revised form 24April 2 Abstract This paper deals with a thermodynamically consistent numerical formulation for coupled thermoplastic problems including phase-change phenomena and frictional contact. The nal goal is to get an accurate, ecient and robust numerical model, able for the numerical simulation of indus solidi cation processes. Some of the current issues addressed in the paper are the following. A fractional step method arising from an operator split of the governing di erential equations has been used to solve the nonlinear coupled system of equations, leading to a staggered product formula solution algorithm. Nonlinear stability issues are discussed and isentropic and isothermal operator splits are formulated. Within the isentropic split, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain xed in the mechanical problem. The formulation of the model has been consistently derived within a thermodynamic framework. All the material properties have been considered to be temperature dependent. The constitutive behavior has been de ned by a thermoviscous/ elastoplastic free energy function, including a thermal multiphase change contribution. Plastic response has been modeled by a J2 temperature dependent model, including plastic hardening and thermal softening. The constitutive model proposed accounts for a continuous transition between the initial liquid state, the intermediate mushy state and the nal solid state taking place in a solidi cation process. In particular, a pure viscous deviatoric model has been used at the initial uid-like state. A thermomecanical contact model, including a frictional hardening and temperature dependent coupled potential, is derived within a fully consistent thermodinamical theory. The numerical model has been implemented into the computational * Corresponding author. Tel.: fax: address: agelet@cimne.upc.es C. Agelet de Saracibar) /1/$ - see front matter # 21 Elsevier Science Ltd. All rights reserved. PII: S )94-2

2 1566 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 nite element code COMET developed by the authors. Numerical simulations of solidi cation processes show the good performance of the computational model developed. # 21 Elsevier Science Ltd. All rights reserved. Keywords: A. Solidi cation and melting A. Thermomechanical processes B. Constitutive behaviour C. Finite elements C. Numerical algorithms 1. Introduction Numerical solution of coupled problems using staggered algorithms, is an ecient procedure in which the original problem is partitioned into several smaller sub-problems which are solved sequentially. For thermomechanical problems the standard approach exploits a natural partitioning of the problem in a mechanical phase, with the temperature held constant, followed by a thermal phase at xed con guration. As noted in Simo and Miehe 1992) this class of staggered algorithms falls within the class of product formula algorithms arising from an operator split of the governing evolution equations into an isothermal step followed by a heat-conduction step at xed con guration. A recent analysis in Armero and Simo 1992a,b, 1993) shows that this isothermal split does not preserve the contractivity property of the coupled problem of nonlinear) thermoelasticity, leading to staggered schemes that are at best only conditionally stable. Armero and Simo 1992a,b, 1993) proposed an alternative operator split, henceforth referred to as the isentropic split, whereby the problem is partitioned into an isentropic mechanical phase, with total entropy held constant, followed by a thermal phase at xed con guration. It was shown by Armero and Simo 1992a,b, 1993) that such an operator split leads to an unconditionally stable staggered algorithm, which preserves the crucial properties of the coupled problem. The aim of the paper is to extend the formulation given by Armero and Simo 1992a,b, 1993) and Simo 1994) to coupled thermoplastic problems with phase-change, to get an accurate, ecient and robust numerical model, able for the numerical simulation of solidi cation processes in the metal casting industry. The remainder of the paper is as follows. Section 2 deals with the formulation of the local governing equations of the coupled thermoplastic problem, consistently derived within a thermodynamic framework. An additive split of the strain tensor and the entropy has been assumed. The constitutive behavior has been de ned by a thermoviscous-elastoplastic free energy function, with temperature dependent material properties. Latent heat associated to the phase-change phenomena has been incorporated to the thermal contribution of the free energy function. Plastic response has been modeled by a J2 temperature dependent model, including nonlinear hardening due to plastic deformation and thermal linear softening. A thermofrictional contact model is derived within a fully consistent thermodinamical framework. Following Laursen in press), contact temperature and entropy variables at the contact surface are introduced, and the energy balance equation at the contact surface is formulated. A particular thermofrictional contact model including frictional hardening and temperature coupling is outlined. Following an analogy

3 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± with the bulk continua, an additive split of the tangential gap and the contact entropy, into elastic and inelastic parts, has been assumed. A pressure and contact temperature dependent thermal contact model has been used. Additionally, a gap dependent thermal model has been used to take into account surface heat transfer phenomena when the two bodies lose contact. Heat generation due to frictional dissipation is implicitly included. This section ends with the variational formulation of the coupled problem. In Section 3, fractional step methods arising from an operator split of the governing di erential equations are considered. Isentropic and isothermal splits are introduced and nonlinear stability issues linked to the splits are addressed. A key point of the formulation of the isentropic split is the set up of the additional design constraints to de ne the mechanical problem. Here, a strong operator split design constraint is introduced, by requiring that the elastic and plastic entropy, as well as the phase-change induced elastic entropy due to the latent heat, remain xed in the mechanical problem. These additional constraints motivate the de nition of the sets of variables and nonlinear operators introduced in the present formulation. Within the time discrete setting, the additive operator splits lead to a product formula algorithm and to a staggered solution scheme of the coupled problem. Finally, the time discrete variational formulation of the coupled problem, using isentropic and isothermal splits, is introduced. Section 4deals with numerical simulations of solidi cation processes. Some concluding remarks are drawn in Section 5. Finally, a step-by-step formulation of the thermoplastic and thermofrictional return mapping algorithms within the mechanical and thermal problems arising from an isentropic split of their governing equations is given in Appendices A and B. 2. Formulation of the coupled thermoplastic problem We describe below the system of quasi-linear partial di erential equations governing the evolution of the coupled thermomechanical initial boundary value problem, including thermal multiphase change and thermofrictional contact Local governing equations Let 24n dim 43 be the space dimension and I :ˆ TŠ R the time interval of interest. Let the open sets R n dim with smooth and closure :ˆ be the reference placement of a continuum body B. Denote by ' :I! R n dim the orientation preserving deformation map of the body B, with material velocity V t ' ˆ ' :, deformation gradient F :ˆ D' and absolute temperature : I! R. For each time t 2 I, the mapping t 2 I7!' t : ˆ ' t represents a one-parameter family of con gurations indexed by time t, which maps the reference placement of body B onto its current placement S t : ' t B R n dim : The local system of partial di erential equations governing the coupled thermomechanical initial boundary value problem is de ned by the momentum and energy

4 1568 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 balance equations, restricted by the inequalities arising from the second law of the thermodynamics. This system must be supplemented by suitable constitutive equations. Additionally, one must supply suitable prescribed boundary and initial conditions, and consider the equilibrium equations at the contact interfaces Local form of momentum and energy balance equations The local form of the momentum and energy balance equations can be written in a rst order system form as, see, e.g. Truesdell and Noll 1965), ' : 9 V : ˆ V ˆ DIV Š B in I 1 H : ˆ DIV QŠ R D int > where :! R is the reference density, B are the prescribed) body forces per unit reference volume, DIV Šthe reference divergence operator, the Cauchy stress tensor, H the entropy per unit reference volume, Q the nominal) heat ux, R the prescribed) heat source and D int the internal dissipation per unit reference volume. Formally, the governing equations for a quasi-static case, may be obtained just by setting ˆ in 1) Dissipation inequalities The speci c entropy H and the Cauchy stress tensor are de ned via constitutive relations, typically formulated in terms of the internal energy E, and subjected to the following restriction on the internal dissipation, see, e.g. Truesdell and Noll 1965) D int ˆ : : H : E : 5 in I 2 where :ˆ SYMM F IŠ is the in nitesimal strain tensor. Here SYMM Š denotes the symmetric operator and I is the second order identity tensor. The heat ux Q is de ned via constitutive equations, say Fourier's law, subjected to the restriction on the dissipation by conduction given by D con ˆ 1 GRAD ŠQ5 in I Thermoplastic/viscous constitutive equations Micromechanically based phenomenological models of in nitesimal strain plasticity adopt a local additive decomposition of the strain tensor into elastic and plastic parts. Hardening mechanisms in the material taking place at a microstructural level are characterized by an additional set of phenomenological internal variables collectively denoted here by. A viscoelastic behaviour is characterized by an additional strain variable denoted as. The aims behind the introduction of this viscous behaviour are to adopt a pure deviatoric viscous model at very high temperature, i.e. at the liquid state, and to introduce viscous e ects at the mushy/solid state. In the

5 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± coupled thermomechanical theory, an additive split of the local entropy into elastic and plastic parts is adopted, where the plastic entropy is viewed as an additional internal variable arising as a result of dislocation and lattice defect motion. This additive split of the local entropy was adopted by Armero and Simo 1993). The above considerations, motivate the following additive split of the in nitesimal strain tensor :ˆ e E p and local entropy H :ˆ H e H p and the following set of microstructural plastic/viscous internal variables G :ˆ f p H p g. The internal energy function E^ depends on the elastic part of the strain tensor e, the hardening internal variables, the con gurational entropy H e and the viscous strain tensor taking the functional form E ˆ E^ e H e. Introducing the functional form of the internal energy into the expression of the internal dissipation, taking the time derivative, applying the chain rule and using the additive split of the in nitesimal strain tensor and total entropy, a straightforward argument yields the following constitutive equations and reduced internal dissipation inequality :ˆ :ˆ :ˆ ee^ e H H ee^ H e E^ e H e E^ e H e 4 D int :ˆD mech D ther 5 with D mech :ˆ : : p : : : 5 and D ther :ˆ H : p : 5 Using the Legendre transformation ˆ E H e, the free energy function takes the functional form ˆ ^ e : Taking the time derivative of the free energy function and applying the chain rule, a straightforward argument yields the following alternative expressions for the constitutive equations 9 p ^ e H e ^ e 6 ^ e ^ > e Assuming a yield function of the form :ˆ^, the evolution laws of the plastic internal variables, assuming associated ow, take the form ^ :p :ˆ : :ˆ H : p ^ ^ 7

6 157 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 and the following Kuhn±Tucker 5 U 4 ˆ and consistency : ˆ conditions must be satis ed for a rate-independent plastic model. The evolution law for the viscous strain takes the form : :ˆ 1 v 8 where v is the e ective elastic viscosity material property. Additionally, the heat ux is related to the absolute temperature through the Fourier's law, which for the isotropic case takes the form Q ˆK GRAD Š, where K is the temperature dependent thermal conductivity. Remark 1 Equivalent forms of the energy balance equation). Using the additive split of the total entropy into elastic and plastic parts and the additive split of the internal dissipation into mechanical and thermal, the reduced energy equation can be expressed as H : e ˆDIV QŠ R D mech in I: 9 Alternatively, using the constitutive equation of the elastic entropy, taking its time derivative and applying the chain rule, the temperature-form of the reduced energy equation can be written as c : ˆDIV QŠ R D mech H ep in I 1 with c 2 e h H ep 2 ^ H e e : : 2 ^ e 2 ^ e : : i where c is the reference heat capacity and H ep the structural elastoplastic heating. Using the constitutive Eqs. 6) the reference heat capacity and structural elastoplastic heating, take the compact form 11 c :ˆ H ep H^ e : : 12 D mech Remark 2 Thermal phase-change contributions). The free energy function for a coupled thermomechanical model including phase change can be splitted into thermoelastic te thermoplastic tp, thermal except phase change) t and thermal phase change tpc contributions, taking the functional form, ˆ ^ te e ^ tp ^ t ^ tpc :

7 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Collecting into a thermoelastoplastic part ^ tep E e all the terms appearing into the free energy function, except the thermal phase change contribution, and setting H e :ˆ H e tep He tpc with H e tep :ˆ@ ^ tep e and H e tpc :ˆ@ ^ tpc the reduced energy balance equation in entropy form, can be written as H : e tep ˆDIV QŠ R D mech H pc with H pc :ˆ L^ ˆ H : e tpc ˆ@ 2 ^ tpc : 13 where H pc :ˆ L : is the phase-change heating given by the rate of latent heat L per unit reference volume. Similarly, the reference heat capacity can be splitted into c ˆ c tep c tpc where c tep ˆ@ 2 ^ tep e and c tpc ˆ@ 2 ^ tpc and the temperature form of the energy balance equation takes the form c tep : ˆDIV QŠ R D mech H ep H pc with H pc :ˆ L : ˆ c tpc : ˆ@ 2 ^ tpc : : 14 Remark 3 Mechanical modeling of the liquid phase). The constitutive model presented above accounts for thermoviscous, themoelastic and thermoplastic behaviour. In particular, a pure thermoviscous constitutive model may be used at the initial liquid state and a thermoelastoplastic, with or without viscous e ects, at the nal solid state of a solidi cation process, while a continuous transition model between the initial liquid state, the intermediate mushy state and the nal solid state may be de ned in terms of the di erent phase fractions involved, i.e. the liquid fraction f l 2 1Š and the solid fraction f s 2 1Š such that f l ˆ 1 f s. A simple continuous model, able to characterize from a pure viscous deviatoric) behaviour at the liquid state to a pure elastic behaviour at the solid state, can be obtained dividing by f s the shear deformation modulus and dividing by f l the elastic viscous material properties A J2-thermoplastic constitutive model Here the following J2-thermoplastic constitutive model described in Boxes 1±3 is considered. Uncoupled thermoelastic and hardening contributions to the free energy function given in Box 1 are considered, as suggested by experimental results in Zbedel and Lehmann 1987). All thermomechanical material properties are considered to be temperature dependent. A particular interest has been placed in considering the case in which the speci c heat is temperature dependent and the latent heat is non-zero. Note that in this case the functional related to the pure thermal contribution is obtained through an integral expression. Phase-change straining due to shrinkage is considered within the thermoelastic coupling contribution to the free energy function. Mechanical behaviour within the transition between the initial liquid-like state and the nal solid state has been accounted for dividing by the solid

8 1572 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Box 1. J2-thermoplastic constitutive model. Free energy function Free energyfunction ^ e ˆ W^ e M^ e T^ K^ i. Linear hyperelastic response > > W^ e ˆ W^ dev e Š U^ tr e Š W^ dev e Š ˆ dev 2 E e Š U^ tr e Š ˆ 1 2 tr2 e Š with ˆ =f s where f s 2 1Š. ii. Thermoelastic coupling, where M^ e e^ ˆ e^ e^ Štr e Š :ˆ 3 ref e ps iii. Thermal contribution c s > IF c s ˆ constant and L ˆ then T^ ˆ c s log = Š else T^ ˆ T^ d end if iv. Hardening potential, T^ ˆ h o c s K^ ˆ 1 2 h 2 y y 1 ŠH^ 1 exp where H^ :ˆ Š= if 6ˆ if ˆ : L i d

9 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Box 2. J2-thermoplastic constitutive model. Thermoelastic/viscous response Thermoelastic response i. Cauchy stresses, ˆ p1 3 s p ˆ tr e Š e^ e^ Š s ˆ 2 dev e Š: ii. Elastic entropy, else if c s ˆ constant and L ˆ then h i H e ˆ c s log = 3 e ps tr e ŠK^ H e ˆ 3 ref tr E e Š e^ e^ Štr e Š W^ H e h c s 3 H L W^ e i d 3 eps tr e ŠK^ ref tr E e Š e^ e^ Štr e Š end if Thermoviscous response i. Viscous stress, ˆ ii. Evolution equations v > f l 2 1Š dev : 1 ˆ s ˆ fl v s tr : 1 ˆ vol v dev v 3p:

10 1574 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Box 3. J2-thermoplastic constitutive model. thermoplastic response Thermoplastic response i. Von Mises yield function with ow stress Y :ˆ y r ^ q ˆ dev Š 2 Y qš: 3 ii. Hardening variable q conjugate to, q ^ ˆh y y 1 1 exp Š: iii. Linear thermal softening, 9 y ˆ y 1 w Š y 1 ˆ y 1 1 w 1 Š > h ˆ h 1 w h Š: iv. Plastic evolution laws, : p :ˆ n with n :ˆ dev Š= dev Š 9 : p :ˆ 2=3 H : p p :ˆ > 2=3Y : volume fraction f s the given shear modulus and dividing by the liquid volume fraction f l the given elastic viscosity v : 2.3. Thermomechanical contact model Contact kinematics Let 24n dim 43 be the space dimension and I :ˆ TŠ R the time interval of interest. Let the open sets 1 R n dim and 2 R n dim with smooth 1 2 and closures X 1 :ˆ and :ˆ 2 2, be the reference placement of two continuum bodies B 1 and B 2, with material particles 1 labeled X 2 2 and Y 2 respectively. Denote by ' i i : I! R n dim the orientation preserving deformation map of the body B i, with material velocities V i t ' i and deformation gradients F i :ˆ D' i. For each time t 2 I, the mapping t 2 I7!' i :ˆ ' i t represents a t

11 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± one-parameter family of con gurations indexed by time t, which maps the reference placement of body B i onto its current placement S i t : ' t i B i R n dim. We will denote as the contact surface i the part of the boundary of the body B i such that all material points where contact will occur at any time t 2 I are included. The current placement of the contact surface i is given by i :ˆ ' t i i. Attention will be focussed to material points on these surfaces denoted as X 2 1 and Y 2 2. Current placement of these particles is given by x ˆ ' t 1 X 2 1 and y ˆ ' t 2 Y 2 2. Using a standard notation in contact mechanics we will assign to each pair of contact surfaces involved in the problem, the roles of slave and master surface. In particular, let 1 be the slave surface and 2 be the master surface. Additionally, we will denote slave particles and master particles to the material points of the slave and master surfaces, respectively. With this notation in hand, we will require that any slave particle may not penetrate the master surface, at any time t 2 I. Although in the continuum setting the slave-master notation plays no role, in the discrete setting this choice becomes important Contact surfaces parametrization. Let A i R n dim1 be a parent domain for the contact surface of body B i. A parametrization of the contact surface for each body B i is introduced by a family of orientation preserving) one-parameter mappings indexed by time, t : A i R n dim1 i! R n dim such that i i :ˆ A i and i i :ˆ t A i i. Using the mapping composition rule, it also follows that t ˆ ' t i i. It will be assumed in what follows that these parametrizations have the required smoothness conditions. Within the slave-master surface role, focus will be placed on the parametrization of the master surface. Using theparametrization of the contact surfaces introduced above we consider a point :ˆ 1 2 2A 2 of the parent domain, such that 2 2 Y :ˆ y :ˆ t 2 E :ˆ e 15 2 :ˆ t where Y and y are, respectively, the reference and current placement of a master particle and E and e, ˆ 1 2 are the convected surface basis attached to the master particle Y 2 2, on the reference and current con guration, respectively. Here denotes partial derivative with respect to Closest-point-projection map and normal gap. Let Y^ : 2! 1 be the closest-point-projection map de ned as n o Y^ X t :ˆ arg min X ' 2 Y 16 ' t 1 Y2 2 t and set Y : Y^ X t y :ˆ ' 2 Y t 17

12 1576 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 where y 2 2 is the closest-point projection of the current position of the slave point X onto the current placement of the master surface 2. Let g N :ˆ g^n X t be the normal gap function de ned for any slave particle X 2 1 and for any time t 2 I as g N :ˆ g^n X t :ˆ 'ŠŠ 18 where 'ŠŠ :ˆ ' t 1 X ' t 2 Y gives the jump of the deformation map at the contact surface and : 2! S 2 is the unit outward normal eld to the current placement of the master surface particularized at the closest-point projection y Convected basis, metric and curvature tensors at the closest-point projection. Associated to the closest-point projection given by 16), for some point :ˆ 1 2 2A 2 of the parent domain we will have Y :ˆ 2 y :ˆ 2 t. 19 Attached to the master particle Y 2 2 we de ne the convected surface basis on the reference and current con gurations, respectively, as ref :ˆ E :ˆ e 2 Additionally, the unit outward normals ref 2 S 2 and 2 S 2 at the master particle Y on the reference and current con gurations, respectively, can be de ned as ref :ˆ ref 1 ref 2 ref 1 ref :ˆ 1 2 k k 21 The vectors ref 2 T ref S 2 and 2 T S 2, ˆ 1 2 span the tangent spaces T refs 2 and T S 2 to the S 2 unit sphere at j ref and, respectively. Here the tangent space to the S 2 unit sphere at 2 S 2 is de ned as T S 2 :ˆ 2 R n dim f : ˆ g 22 The convected surface basis vectors ref and, ˆ 1 2, augmented with the unit outward normals ref and, provides local spatial frames at the master particle Y X t on the reference and current con gurations, respectively. The convected surface basis vectors ref and, ˆ 1 2, induces a surface metric or rst fundamental form on the reference and current con gurations, de ned respectively as M : ref ref m :ˆ 23

13 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Inverse surface metrics M and m are de ned in the usual manner. Additionally, dual surface basis on the reference and current con gurations are straightforward de ned respectively as ref :ˆ M ref :ˆ m 24 The variation of the convected surface basis along the convected coordinates, together with the unit normal, induces the second fundamental form or surface curvature de ned, on the reference and current con gurations, as ref :ˆ E ref :ˆ e Local form of energy balance equation The local form of the energy balance equation at the contact surface take the form, see Laursen in press), c H : c ˆ Q c 1 Q c 2 D cint on 1 I 26 where c > is the absolute contact temperature, H c the contact entropy per unit reference surface 1, Q c, ˆ 1 2 are the outward normal nominal heat uxes on the contact surface of bodies B, ˆ 1 2, per unit reference surface 1 and D cint is the contact internal dissipation per unit reference surface Dissipation inequalities The contact speci c entropy H c is de ned via constitutive relations, typically formulated in terms of the contact internal energy E c, and subjected to the following restriction on the contact internal dissipation, see Laursen in press),. D cint :ˆt 1 VŠŠ c H : c E : c4 on 1 I 27 where VŠŠ :ˆ V 1 V 2 denotes the jump in the material velocities eld across the contact surface and t 1 is the nominal contact traction on 1. Introducing the nominal contact pressure t N, as the projection of the nominal traction t 1 onto the unit normal, and nominal frictional tangent traction components t T on the convected dual basis, ˆ 1 2, as minus) the projection of the nominal traction t 1 onto the convected basis, t N :ˆ t 1 t T :ˆ t [ 1 28 and using the expressions g : N ˆ VŠŠ g : H j gnˆ:ˆ VŠŠ 29

14 1578 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 and the following relation holds t 1 VŠŠ ˆ t N g : N t T g : T 3 the contact internal dissipation given by 27) takes the form D cint :ˆ t N g : N t T g : T c H : c E : c5 on G 1 I 31 The outward normal nominal heat uxes Q c, ˆ 1 2, are subjected to the following restriction on the dissipation by conduction D ccon :ˆ Q c 1 Q 2 c 1 1 c 5 on 1 I c Thermofrictional contact constitutive equations Phenomenological frictional models adopt a local additive decomposition of the tangential gap into elastic and plastic parts. Frictional hardening mechanisms taking place at a microstructural level are characterized by an additional set of phenomenological internal variables collectively denoted here by c. Furthermore, an additive split of the local contact entropy into elastic and plastic has been adopted, following a parallel approach to the bulk continua case. The above considerations, motivate the following additive split of the tangential gap g T :ˆ g e T gp T, ˆ 1 2, and contact local entropy per unit reference n surface Ho c :ˆ H e c Hp c and the following set of frictional internal variables G p c :ˆ gp T H p c c. The internal contact energy function E^ c per unit contact surface 1 depends on the normal gap, the elastic part of the tangential slip g e T, the frictional hardening internal variables c and the con gurational contact entropy H e c, taking the functional form E c ˆ E^ c g e T He c c. Introducing the functional form of the contact internal energy into the expression of the contact internal dissipation, taking the time derivative, applying the chain rule and using the additive split of the tangential gap and total contact entropy, a straightforward argument yields the following contact constitutive equations and reduced internal dissipation inequality t N gn E^ c g N g e T He c 9 c t T g e T E^ c g N g e T He c c c H e c E^ g N g e T He c c q c :ˆ@ c E^ c g N g e T He c > c D cint :ˆD cmech D cther 5 33 with D cmech :ˆ t T g : p T q c : c5 and D cther :ˆ c H : p c : 34

15 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Using the Legendre transformation c ˆ E c c H e c, the contact free energy function takes the functional form c ˆ ^ c g N g e T c c. Taking the time derivative of the contact free energy function and applying the chain rule, a straightforward argument yields the following alternative expressions for the contact constitutive equations t N gn ^ c g N g e T 9 c c t T g e ^ T g N g e T c c H e c ˆ@ c ^ c g N g e T c c q c :ˆ@ c ^ c g N g e T > c c : 35 Constitutive equations for the nominal) normal heat conduction uxes per unit reference contact surface 1 are given by Q 1 ccond ˆ h^ 1 cond t N c g 1 with g 1 :ˆ 1 c Q 2 ccond ˆ h^ 2 cond t N c g 2 with g 2 ˆ 2 c 36 where h^ cond t N c >, ˆ 1 2, is the heat transfer coecient of the surface, ˆ 1 2, assumed to be a function of the contact pressure and the contact temperature. Note that 36) allows that the contact dissipation by heat conduction restriction given by 32) is unconditionally satis ed. Assuming a slip function of the form c ˆ ^ c t N t T c q c, the evolution laws of the plastic internal variables take the form g : p T ˆ 9 c@ tt ^ c t N t T c q c H : p c ˆ c ^ c t N t T c q c 37 : c ˆ q c ^ c t N t T c q c > and the following Kuhn±Tucker c 5, c 4, c c ˆ and consistency c : c ˆ conditions must be satis ed for a rate-independent frictional model. Remark 4 Equivalent forms of the energy balance equation). Using the additive split of the total contact entropy into elastic and plastic parts and the additive split of the contact internal dissipation into mechanical and thermal, the reduced contact energy equation can be expressed as c H : e c ˆ Q c 1 Q c 2 D cmech on G 1 I 38

16 158 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Alternatively, using the constitutive equation of the contact elastic entropy, taking its time derivative and applying the chain rule, the temperature-form of the reduced contact energy equation can be written as c c : c ˆ Q c 1 Q c 2 D cmech H fc on 1 I 39 where the reference contact heat capacity c c and the frictional contact heating H fc take the form c c :ˆ 2 ^ c c c g N g e T c c 4 h H fc : ˆ 2 ^ c g N c g N g e T : c c g 2 ^ c g e c g N g e T : c c g e T 2 ^ c c c g N g e T : c c c i 41 or, alternatively, using the constitutive Eq. 35), c co :ˆ c H e c Hfc :ˆ c t n g : n t T g : T D cmech : A Thermofrictional constitutive model Here the following thermofrictional constitutive model described in Boxes 4and 5 is considered. Coupled thermofrictional behaviour is considered within the thermofrictional hardening potential. Box 4. Thermofrictional constitutive model. Free energy function Free energyfunction ^ c g N g e T c c ˆ W^ c g N g e T T^ c c K^ c c c i. Linear hyperelastic response N >, T > ), W^ c g N g e 1 T :ˆ N g N 2 Tg e T M g e T : ii. Thermal contribution c c >, T^ c c :ˆ c c c c log c = Š: iii. Hardening potential, K^ c c c :ˆ Xm 1 n 1 ^ n c c : nˆ1

17 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Box 5. Thermofrictional constitutive model. Thermofrictional response Thermoelastic response i. Contact pressure and frictional traction. t N ˆ N g N t T ˆ T M g e T ii. Elastic entropy H e c ˆ c clog c = ˆ Xm ^ n c c : nˆ1 Thermoplastic response i. Slip function ^ c t N t T c q c :ˆ t [ T ref ^ c q c t N where t [ T ref :ˆ t T M 1=2: t T ii. Hardening variable q c conjugate to c, q c :ˆ@ c ^ c ˆ Xm nˆ1 ^ n c c n : iii. Linear thermal softening, ^ n c ˆ ^ n 1 w n c Š n ˆ 1 m: iv. Frictional mechanical dissipation, D cmech ˆ c ^ c t N :

18 1582 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Variational formulation Using standard procedures, the weak form of the momentum balance we will assume the quasi-static case for simplicity) and reduced energy balance equations in I take the following expressions: D E D E GRAD B t t t 2 2 D H : E e D E Q GRAD Š h R Dmech i Q Q D E D E QN QN 2 1 ˆ 2 ˆ 2 which must hold for any admissible displacement and temperature variations and, respectively. Here hidenotes the L 2 -inner product and witha slight abuse in notation hi, hi Q and hi denote the L 2, L 2 Q and L2 -inner products on the boundaries, Q and, respectively. For the sake of notation compactness, it is implicitly assumed that :ˆ [ 2 ˆ1, :ˆ[ 2 ˆ1 and Q :ˆ[ 2 ˆ1 Q. Using the equilibrium of forces at the contact interface and introducing the nominal) contact heat uxes Q c 1 and Q c 2 per unit reference surface 1, the following expressions hold D E D E t 1 2 t ˆ 2 QN 1 o 1 Q 1 1 c o 1 ˆ D E D E 1 QN Q 2 2 c 2 ˆ 1 43 Using 44) the weak form of the balance Eqs. 43) take the form: D E GRAD B t t ˆ D 1 H : E e D E D Q GRAD Š h R Dmech i Q Q 1 Q D E Q c 2 2 ˆ 1 c 1 E 1 45 where it is pointed out that all contact contributions are parametrized in terms of the contact surface 1. Similarly, the weak form of the reduced energy balance equation on 1 I take the form: D c H : E D E e c c Q c 1 Q 2 2 c D cmech c ˆ

19 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± which must hold for any admissible contact temperature variation c. Adding the weak form of the energy balance on 1 I given by 46), to the weak form of the energy balance in I given by 45) 2, the following alternative expressions are obtained D E GRAD B t t ˆ D 1 H : E e D E Q GRAD Š h R Dmech i Q Q c D E D Q c 2 2 c ch : E 47 e 1 c c D cmech c 1 ˆ 1 3. Time integration of the coupled thermoplastic problem The numerical solution of the coupled thermomechanical IBVP involves the transformation of an in nite dimensional dynamical system, governed by a system of quasi-linear partial di erential equations into a sequence of discrete nonlinear algebraic problems by means of a Galerkin nite element projection and a time marching scheme for the advancement of the primary nodal variables, i.e. displacements and temperatures, together with a return mapping algorithm for the advancement of the internal variables. Here, attention will be placed to the time integration schemes of the governing equations of the coupled thermoplastic problem. In particular, we are interested in a class of unconditionally stable staggered solution schemes, based on a product formula algorithm arising from an operator split of the governing evolution equations. These methods fall within the classical fractional step methods Local evolution problem Consider the following homogeneous) rst order dissipative local problems of evolution, see Simo 1994), Agelet de Saracibar et al. 1997a,b,c), Laursen in press), i. Evolution problem in I. Consider the local dissipative problem of evolution for the homogeneous) thermoplastic problem, written as 9 d dt Z ˆ AZ CŠ in I 48 > Zj tˆoˆ Z in along with 9 d dt Cp ˆ G p Z CŠ in I > and C p j tˆ in d dt Cv ˆ G v Z CŠ in I C v j tˆˆ in 9 > 49

20 1584 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 where Z, lying in a suitable Sobolev space Z, is a set of primary independent variables, C :ˆ fc p C v g is a set of internal variables, AZ CŠ and GZ CŠ are nonlinear operators and 5 is a plastic multiplier. ii. Evolution problem on 1 I. Consider the dissipative local problem of evolution for the frictional contact problem, written as d h i 9 dt Z c ˆ A c Z C on 1 I 5 > Z c j tˆ Z c on 1 along with d h i 9 dt C c ˆ c G c Z C on 1 I > C c j tˆˆ on 1 51 where Z c, lying in a suitable Sobolev space Z c, is a set of primary independent variables, C c is a set of internal variables, A c Z C Š and G c Z C Š are nonlinear operators and c 5 is a frictional multiplier. Additionally, collecting variables and using a compact notation, we introduce the following set of variables, Z :ˆ fz Z c g and C :ˆ fc C c g, and nonlinear operators, A Z C Š :ˆ fa Z CŠ A c Z CŠg and G Z C Š :ˆ fgz CŠ G c Z C Šg. In the formulation of the fractional step method described below, it is essential to regard the set of internal variables C as implicitly de ned in terms of the variables Z via the evolution equations. Therefore Z are the only independent variables and their choice becomes a crucial aspect in the formulation of the fractional step method. We refer to Simo 1994), Agelet de Saracibar et al. 1997a,b,c) and Laursen in press) for further details. Motivated by the structure and the design constraints of the isentropic split described below, we consider the following set of conservation/entropy/latent heat variables Z Z :ˆ fz Z c g where Z :ˆ ' p H e H p L Z c :ˆ H e c Hp c in I on 1 I along with the set of plastic strain/hardening/viscous and frictional slip/hardening variables C de ned as C :ˆ fc C c gwith C :ˆ fc p C v g 54

21 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± where C p :ˆ f p gand C v :ˆ dev Š tr Š C c :ˆ g p T c in I on 1 I 55 Here ' is the deformation map, p :ˆ V denotes the material linear momentum, H e and H p are the elastic entropy including phase change contributions) and plastic entropy per unit reference volume, respectively, L is the latent heat per unit reference volume, p the plastic strain, the strain-like hardening variables and the viscous strains. Additionally, H e c and Hp c are the elastic and plastic contact entropy per unit reference surface, g p T the plastic slip and c the frictional hardening variables. All the remaining variables in the problem can be de ned in terms of Z and C by kinematic and constitutive equations. In particular, 1. The elastic strain e :ˆ p. 2. The Cauchy stress tensor ee^ e H e and stress-like hardening variables :ˆ@ E^ e H e. 3. The temperature H ee^ e H e and nominal heat ux Q ˆKGRAD Š. 4. The normal gap g N :ˆ 'ŠŠ and tangential gap g T given by the time integration of g : T :ˆ VŠŠ. 5. The nominal contact pressure t N gn E^ c g N g e T He c c, nominal frictional traction t T g e T E^ c g n g e T He c c and conjugate of the frictional hardening variable q c :ˆ@ c E^ c g N g e T He c c. 6. The contact temperature c H e c E^ c g N g e T He c c Thermoplastic/viscous nonlinear operators in I With these de nitions in hand, and assuming zero body forces and zero heat sources, the governing evolution equations of the thermoplastic/viscous problem can be written in the form given by 48) and 49) where the thermoplastic/viscous nonlinear operators AZ CŠ and GZ CŠ ˆ G p Z CŠ G v Z CŠ take the form AZ Š :ˆ 8 >< 1 p DIV Š 1 DIV QŠ 1 D mech 9 1 D ther >: H ) q G p Z Š :ˆ G v Z CŠ q ^ q > 8 >< >: 1 dev v 1 vol v 9 dev Š tr Š > 56

22 1586 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 where the phase-change heating H pc is given by H pc :ˆ@ 2 ^ tpc : 57 and the mechanical dissipation D mech and thermal dissipation D ther are given by D mech :ˆ R p : G p Z CŠ R v : G v Z CŠ5 D ther ^ 58 where, using a compact notation, we have denoted as R pt :ˆ T and R v T :ˆ dev Š T 1 3 tr Š the generalized plastic and viscous stress tensors, respectively Thermofrictional nonlinear operators on 1 I With the above de nitions in hand the governing evolution equations of the thermofrictional problem can be written in the form given by 5) and 51) where the thermofrictional nonlinear operators A c Z C Š and G c Z C Š take the form 8 1 h i Q c 1 Q c >< D c:mech c c A c Z C :ˆ 1 >: D cther > c h i G c Z C :ˆ 8 < tt ^ c t N t T c q q c ^ c t N t T c q c 9 = 59 where the contact mechanical dissipation D cmech and thermal dissipation D cther are given by D cmech : t T g : p T q c : c > D cther :ˆ c c ^ c t N t T c q c : A-priori stability estimate For nonlinear dissipative problems of evolution nonlinear stability can be phrased in terms of an a-priori estimate on the dynamics of the form d dt L Z C 4 for t 2 TŠ 61 where L is a non-negative Lyapunov-like function.

23 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± For nonlinear thermoplasticity featuring frictional contact behaviour, see Armero and Simo 1993) and Laursen in press), consider an extended canonical free energy functional L de ned as " p # 2 L Z C ˆ E^ e H e H e d V ext ' 2 62 h E^ c g N g e T He c i c H e c d 1 and assume that the following conditions hold, 1. Zero heat sources, i.e. R, 2. Conservative mechanical loading with potential V ext ', i.e. DV ext :ˆ B t 3. Dirichlet boundary conditions for the temperature eld with prescribed constant temperature >, i.e. ˆ on I and Q ˆ, or Von-Neuman boundary conditions for the heat ux with zero heat ux, i.e. Q ˆ on Q I and ˆ, or in general mixed boundary conditions satisfying Q ˆ on [ Q I. Then, L is a non-increasing Lyapunov-like function along the ow generated by the thermoplastic problem and a straightforward computation shows that the following a-priori stability estimate holds in I: d dt L Z C ˆ D mech D con Š d 1 c D cmech D ccon d4: 63 This condition is regarded, see Armero and Simo 1993) and Laursen in press), as a fundamental a-priori stability estimate for the thermoplastic problem of evolution which must be preserved by the time-stepping algorithm Operator splits Consider the dissipative problem of evolution given by 48)±51) with the associated non-increasing Lyapunov-like function L given by 62). Consider an additive operator split of the vector eld A ˆ A 1 A 2, with A :ˆ A A c, leading to the following two sub-problems Problem 1 Problem 2 : Z ˆ A h i 9 : 1 Z C Z ˆ A h i 9 2 Z C C : P ˆ G p Z CŠ C : C : p ˆ G p Z CŠ v ˆ G v Z CŠ C : v ˆ G v Z CŠ C : h i c ˆ c G c Z C > C : h i c ˆ c G c Z C : > 64

24 1588 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 The critical restriction on the design of the operator split is that each one of the subproblems must preserve the underlying dissipative structure of the original problem, i.e. d dt L Z 4ˆ 1 2 with t7! Z denotes the ow generated by the vector eld A ˆ 1 2. Two di erent operator splits will be considered here. First, following Armero and Simo 1992a,b, 1993), see also Laursen in press) for frictional contact problems, an isentropic operator split, which satis es the critical design restriction mentioned above, is considered. This split is compared next with an isothermal operator split, which does not satisfy the design restriction The isentropic operator split Consider the following additive isentropic-based operator split of the vector eld A Z Š: A h i Z :ˆ A h i 1 Z A h i 2 Z 66 ise where we de ne the vector elds A n 1 ise :ˆ A 1 ise A 1 A 1 ise ise 65 o n o cise and A 2 ise :ˆ A 2 ise A 2 cise as p >< DIV Š >< Z A 2 isc Z 1 DIV QŠ D mech 1 >: > D ther >: > 67 H pc h i cise Z A 1 :ˆ 8 1 h i A 2 cise Z :ˆ Q c 1 Q c >< D cmech c >: 1 D > cther 68 and consider the following two problems of evolution: Problem 1 Problem 2 : Z ˆ A h i 9 : 1 ise Z Z ˆ A h i 9 2 : ise Z p ˆ G p Z Š : p ˆ G p Z Š : v ˆ G v Z Š : v ˆ G v Z Š : h i c ˆ c G c Z > : h i c ˆ c G c Z : > 69

25 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Within this operator split, Problem 1 de nes a mechanical phase at xed entropy and Problem 2 de nes a thermal phase at xed con guration. Note that a strong condition has been placed in the Problem 1, by the additional requirement that not only the total speci c entropy must remain xed, but also the elastic and plastic entropy, as well as the latent heat. Note also that the evolution of the plastic internal variables is imposed in both problems. Denoting by t7! Z the ow generated by the vector eld A ise, ˆ 1 2, a straightforward computation shows that the following stability estimates hold: d dt L Z 1 1 d dt L Z 2 2 ˆ D 1 mech d 1 ˆ h 2 D 2 mech D2 con 1 c 2 h D 1 cmech d4 i d D 2 cmech D2 ccon i d4 7 where D mech, D con and are the mechanical dissipation, thermal heat conduction dissipation and absolute temperature, respectively, and D cmech, D ccon and c are the contact frictional mechanical dissipation, contact heat conduction dissipation and contact temperature, respectively, in Problem, ˆ 1 2. Thus, the isentropic split preserves the underlying dissipative structure of the original problem The isothermal operator split Consider the following additive isothermal-based operator split of the vector eld A Z Š: A h i Z :ˆ A h i 1 iso Z A h i 2 iso Z 71 where we de ne the vector elds A 1 iso :ˆ A 1 iso A 1 A 1 iso 8 1 n ciso DIV Š >< >< Z Š :ˆ 1 A 2 Hep iso Z Š :ˆ >: 9 > 8 o and A n 2 iso :ˆ A 2 iso A 2 ciso o as 1 DIV QŠ 1 D mech H ep 1 D ther >: H pc 9 > 72

26 159 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± h i < 1 ciso Z H fc = :ˆ c : 8 1 h i >< Q c 1 Q 2 1 c ciso Z c :ˆ 1 >: D cther c A 1 A 2 c D cmech H fc 9 > 73 and consider the following two problems of evolution: Problem 1 Problem 2 : Z ˆ A h i 9 : 1 iso Z C : Z ˆ A h i 9 2 iso Z p ˆ G p Z Š C : C : p ˆ G p Z Š v ˆ G v Z Š C : v ˆ G v Z Š C : h i c ˆ c G c Z > C : h i c ˆ c G c Z : > 74 Within this operator split, Problem 1 de nes a mechanical phase at xed temperature and Problem 2 de nes a thermal phase at xed con guration. Note also, that the evolution of the plastic internal variables is imposed in both problems. Denoting by t7! Z C the ow generated by the vector eld A iso, ˆ 1 2, a straightforward computation shows that the following stability estimates hold: d dt L Z 1 1 C ˆ 1 D 1 mech d D H ep 1 d! cmech d 1 1 c 1 H fc 1 d4= d dt L Z 2 2 C ˆ h 2 D 2 mech D2 con 1 c 2 h i d D 2 cmech D2 ccon c! i d H fc 2 d4= 1 2 H ep 2 d 75 where D mech, D con, Hep and are the mechanical dissipation, thermal heat conduction dissipation, structural elastoplastic heating and absolute temperature,

27 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± respectively, and D cmesh, D ccon, Hfc and c are the contact frictional mechanical dissipation, contact heat conduction dissipation, frictional hardening heating and contact temperature, respectively, in Problem, ˆ 1 2. The contribution of the structural elastoplastic heating and frictional hardening heating to the evolution equations of each one of the problems arising from the isothermal operator split, breaks the underlying dissipative structure of the original problem Product formula algorithms The additive operator split of the governing evolution equations leads to a product formula algorithm and to a staggered solution scheme of the coupled problem, in which each one of the subproblems is solved sequentially. Remember that the set of internal variables C is viewed as implicitly de ned in terms of the set of variables Z, which are considered to be the only independent variables. Therefore, our interest here is placed on the time discrete version of the evolution Eqs. 48) and 5), and the update in time of the variables Z using a time-stepping algorithm. Consider algorithms K t Š being consistent with the ows t7! Z C, ˆ 1 2, and dissipative stable, i.e. which inherit the a-priori stability estimate on the dynamics given by 63). Then the algorithm de ned by the product formula: K t Šˆ K 2 t K 1 t Š 76 is also consistent and dissipative stable. For dissipative dynamical systems if each of the algorithms is unconditionally dissipative stable, then the product formula algorithm is also unconditionally dissipative stable. This product formula algorithm is only rst order accurate. A second order accurate product formula algorithm can be de ned through a double pass technique given by, see Strang 1969), h i K t Šˆ K 1 t=2 K 2 t K 1 t=2 Š: 77 Note that, according to 75), algorithms based on the isothermal operator split will result in staggered schemes at best only conditionally stables and only an isentropic operator split leads to unconditionally dissipative) stable product formula algorithms Time discrete variational formulation The use of an operator split, applied to the coupled system of nonlinear ordinary di erential equations, and a product formula algorithm, leads to a staggered algorithm in which each one of the subproblems de ned by the partition is solved sequentially, within the framework of classical fractional step methods. We note that contrary to common practice, the evolution equations for the microstructural internal variables are enforced in both phases of the operator split, as in Armero and Simo 1992b, 1993) and Simo 1994). A Backward±Euler BE) time stepping algorithm has been used and two di erent operator splits have been considered.

28 1592 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Isentropic split In the isentropic split, rst introduced by Armero and Simo 1992a,b, 1993), the coupled problem is partitioned into a mechanical phase at constant entropy, followed by a thermal phase at xed con guration, leading to an unconditionally stable staggered scheme. Additional design constraints of constant elastic and plastic entropy and latent heat in and constant elastic and plastic contact entropy on 1 have been introduced in the isentropic mechanical phase. An ecient implementation of the split can be done using the temperature as primary variable. See Armero and Simo 1992a,b, 1993) and Simo 1994) for further details Mechanical phase. The mechanical problem is solved at constant entropy. For the sake of simplicity, only the quasi-static case will be considered here. According to the de nition of Z given by 52)±53) and the operator split given by 66)±69), the additional design constraints of constant elastic and plastic entropy and latent heat in and constant contact elastic and plastic entropy on 1, have been introduced. The evolution of the temperature and contact temperature can be computed locally. The time discrete weak form of the momentum balance, local updates of elastic and plastic entropy, latent heat and internal variables in and elastic and plastic contact entropy and internal variables on 1, take the form: D E 9 ~ GRAD B t t~ 1 H~ e ˆ H e n H~ p ˆ Hp n L~ ˆ L n 1 2 ˆ 1 in C~ p ˆ Cp n ~ G ~ p C~ v ˆ C v n tg~ v 9 e H~ c ˆ H e c n p H~ c ˆ H p cn on 1 C~ c ˆ C cn ~ c G ~ > c > 78a 78b and the temperature and contact temperature are locally updated according to ~ H ee^ ~ e He n ~ ~ and ~ c H e c E^ c g~ N g~ e T H e c n ~ c, respectively Thermal phase. Using a BE scheme the time discrete weak form of the energy balance equation, updated elastic and plastic entropy, latent heat and internal

29 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± variables in and elastic and plastic contact entropy and contact internal variables on 1, in the thermal phase, take the form: 9 1 t H e He n Q GRAD Š R Dmech D E D E D E Q Q c 1 1 c Q 1 Q c D E t c H e c H e cn c D cmech c ˆ 1 1 H e ˆ@ ^ e H p ˆ Hp n t D ther L ˆ L C p ˆ Cp n G p C v ˆ Cv n tgv in > 79a H e c ˆ H e c n t Q c 1 Q 2 c H p c ˆ H p cn t D cther c C c ˆ C cn c G c c 9 D cmech c on 1 t where c H e c E^ c g N G e T H e c c. > 79b Isothermal split In the isothermal split the coupled system of equations is partitioned into a mechanical phase at constant temperature in and constant contact temperature on 1, followed by a thermal phase at xed con guration. Note that, within the context of the product formula algorithm, using the entropy form of the energy equation, the elastic entropy and elastic contact entropy computed at the end of the mechanical partition is used as initial condition for the solution of the thermal partition Mechanical phase. Noting that under isothermal conditions the plastic entropy, latent heat and plastic contact entropy remain constant, the time discrete weak form of the momentum balance equation, updated elastic and plastic entropy,

30 1594 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 latent heat and internal variables in and updated elastic and plastic contact entropy and contact internal variables on 1 take the form: D E ~ GRAD B t t~ ˆ 1 e H~ ˆ@ ^ ~ e n~ ~ H~ p ˆ Hp n in L~ ˆ L n C~ p ˆ Cp n ~ G ~ p C~ v ˆ C v n tg~ v 9 e H~ c ˆ@ c ^ c g~ N g~ e T cn ~ c p H~ c ˆ H p c n on 1 C~ c ˆ C cn ~ c G ~ > c : > 8a 8b It is pointed out that the design conditions ~ ˆ n and ~ c ˆ cn have been introduced in 8a) 2 and 8b) 1, respectively Thermal phase. Using a BE scheme the time discrete weak form of the energy balance equation, updated elastic and plastic entropy, latent heat and internal variables in and updated elastic and plastic contact entropy and contact internal variables on 1 take the form: 1 D E t H e H ~ e 9 Q GRAD Š D E D E R D mech Q Q c 1 1 c Q 1 D E Q c D E c 1 t c H e c H~ e c c 1 D cmech c ˆ 1 H e ˆ@ ^ e in 81a H p ˆ Hp n t D ther L ˆ L H C p ˆ Cp n G p C v ˆ Cv n tgv >

31 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± H e e c ˆ H~ c t Q c 1 Q 2 c c t c D cmech H fc 9 H p c ˆ H p c n T c D cther on 1 C c ˆ C cn c G c : > 81b where y c H e c E^ c g N g e T H e c c : 4. Numerical simulations The formulation presented in the previous Sections is illustrated here in a number of representative numerical simulations. The goals are to provide a practical accuracy assessment of the thermomechanical model and to demonstrate the robustness of the overall coupled thermomechanical formulation in a number of solidi cation Fig. 1. Solidi cation of an aluminium ring into a steel mould. Initial geometry of the part and the mould.

32 1596 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 2. Solidi cation of an aluminium ring into an steel mould. a) Four-noded axisymmteric nite element meshes for the part and the mould at the initial con guration b) deformed nite element mesh at 9 s. Fig. 3. Solidi cation of an aluminium ring into an steel mould. Finite element mesh at a deformed con guration of t=9 s. Detail of the deformed mesh at the corner areas.

33 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± examples, including indus processes. The computations are performed with the nite element code COMET developed by the authors. The Newton±Raphson method, combined with a line search optimization procedure, is used to solve the nonlinear system of equations arising from the spatial and temporal discretization of the weak form of the momentum and reduced dissipation balance equations. Convergence of the incremental iterative solution procedure was monitored by requiring a tolerance of.1% in the residual based error norm Solidi cation of an aluminium ring into an steel mould This example deals with the numerical simulation of the solidi cation of an aluminium ring into an steel mould. Fig. 1 shows the initial geometry of the part and the mould. Assumed starting conditions in the numerical simulation of the casting process are given by a completely lled mould with aluminium in liquid state at uniform temperature. The initial temperatures of the aluminium part and the steel mould were 7 and 2 C, respectively. Only gravitational forces have been assumed. The aluminium used for the part presents a sharp liquid±solid phase change between 659 and 66 C. Fig. 4. Solidi cation of an aluminium ring into a steel mould. Temperature distribution at di erent time steps.

34 1598 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 2a shows the four-noded axisymmetric nite element meshes, for both the part and the mould, used in the analysis. Fig. 2b shows the deformed con guration at 9 s of the analysis. A detail of the deformed mesh at the corner areas is shown in Fig. 3. Figs. 4±6 show the distribution of temperature, solid volume fraction and von Mises e ective stress, at di erent time steps. Fig. 7 shows the location of di erent selected points of the part and the mould, where the time evolution of di erent variables has been monitored. Figs. 8 and 9 show the time evolution of the temperature and radial and vertical displacements, respectively, at the selected points shown in Fig Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould This example deals with the numerical simulation of the solidi cation of an Al± 3Mg Si alloy driving wheel into a green sand mould. Fig. 1 shows the geometry of the part. Assumed starting conditions in the numerical simulation of the casting process are given by a completely lled mould with an Al±3Mg Si alloy in liquid state at uniform temperature. The initial temperatures of the alloy part and the Fig. 5. Solidi cation of an aluminium ring into a steel mould. Solid volume fraction distribution at di erent time steps.

35 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 6. Solidi cation of an aluminium ring into a steel mould. Von Mises e ective stress distribution at di erent time steps. Fig. 7. Solidi cation of an aluminium ring into a steel mould. Location of the selected points at the part and the mould.

36 16 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 8. Solidi cation of an aluminium ring into a steel mould. Time evolution of the temperature at some selected points of the part and the mould.

37 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 9. Solidi cation of an aluminium ring into a steel mould. Time evolution of the radial and vertical displacement at some selected points of the part and the mould.

38 162 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 9 continued).

39 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 1. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Geometry of the part. Fig. 11. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Di erent views of the nite element mesh used for the part. Fig. 12. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Location of the selected points at the part and the mould to plot the time evolution of selected results.

40 164 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 green sand mould were 7 and 2 C, respectively. Only gravitational forces have been assumed. The Al±3Mg Si alloy used for the part presents a liquid-solid phase change between 59 and 64 C. Due to the geometry of the wheel, only a 1=5 section of the part and the mould has been discretized, using around 25, tetrahedral elements and 64 nodes. Fig. 11 shows a view of the nite element mesh used in the analysis. The analysis has been carried out in 56 time steps of 15 s. each. Fig. 12 shows the location of di erent selected points to plot time evolution of some results. Fig. 13 shows the temperature evolution at these selected points of the part and the mould. A typical temperature plateau due to the release of latent heat during solidi cation can be observed in these gures. It can also be shown that the points located at the core points D, E and F), take more time to solidify, up to 225 s Fig. 13. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Temperature time evolution at selected points of the part and the mould.

41 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 13 continued). approximately, than the points located at the rim and at the external part points A, B, J and K), which take up to 125 s approximately. Fig. 14shows the location of the sections AA' and BB' selected to show the plot contours of di erent results at various time steps. Fig. 15 shows the solidi cation time on sections AA' and BB'. Figs. 16 and 17 show the temperature distribution on sections AA' and BB', respectively, at di erent time steps. Figs. 18 and 19 show the solid volume fraction distribution on sections AA' and BB', respectively, at di erent time steps. It can be shown that the mushy zone occupies a vast area and there is not any internal area with metal in liquid state. This is due to a very high initial cooling rate and a large di erence between the solidus and liquidus temperatures. Figs. 2 and 21 show the distribution of the von Mises equivalent stress on sections AA' and BB' respectively, at di erent time steps.

42 166 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 14. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Location of the sections AA' and BB' selected to present plot contours of di erent results at various time steps. Fig. 15. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. a) Solidi cation time on section AA' b) solidi cation time on section BB'.

43 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 16. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Temperature distribution on section AA' at di erent time steps. Fig. 17. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Temperature distribution on section BB' at di erent time steps.

44 168 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565±1622 Fig. 18. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Solid volume fraction distribution on section AA' at di erent time steps. Fig. 19. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Solid volume fraction distribution on section BB' at di erent time steps.

45 C. Agelet de Saracibar et al. / International Journal of Plasticity 17 21) 1565± Fig. 2. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Von Mises stress distribution on section AA' at di erent time steps. Fig. 21. Solidi cation of an Al±3Mg Si alloy driving wheel into a green sand mould. Von Mises stress distribution on section BB' at di erent time steps.