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    Acta Materialia 61 (2013) 22682280

    Simulation of solid deformation during solidification: Shearingand compression of polycrystalline structures

    M. Yamaguchi, C. Beckermann

    Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA 52242, USA

    Received 24 October 2012; received in revised form 27 December 2012; accepted 31 December 2012Available online 4 February 2013


    Deformation of the semi-solid mush during solidification is a common phenomenon in metal casting. At relatively high fractions ofsolid, grain boundaries play a key role in determining the mechanical behavior of solidifying structures, but little is known about theinterplay between solidification and deformation. In the present study, a polycrystalline phase-field model is combined with a materialpoint method stress analysis to numerically simulate the coupled solidification and elasto-viscoplastic deformation behavior of a puresubstance in two dimensions. It is shown that shearing of a semi-solid structure occurs primarily in relatively narrow bands near or insidethe grain boundaries or in the thin junctions between different dendrite arms. The deformations can cause the formation of low-angle tiltgrain boundaries inside individual dendrite arms. In addition, grain boundaries form when different arms of a deformed single dendriteimpinge. During compression of a high-solid fraction dendritic structure, the deformations are limited to a relatively thin layer along thecompressing boundary. The compression causes consolidation of this layer into a fully solid structure that consists of numerous sub-grains. It is recommended that an improved model be developed for the variation of the mechanical properties inside grain boundaries. 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

    Keywords: Polycrystalline solidification; Viscoplastic deformation; Phase-field method; Material point method; Grain boundaries

    1. Introduction

    Deformation of the semi-solid mush is a common phe-nomenon in solidifying metal castings. It can lead todefects, such as hot tears, macrosegregation and porosity[1]. Therefore, understanding the mechanical behavior ofthe mush during solidification of metal alloys is of greatimportance in casting simulations incorporating a stressanalysis [2]. In the first part of the present study [3], amodel was developed to simulate the coupled solidificationand deformation of a single dendrite of a pure substance intwo dimensions. The phase-field method [4,5] was used tomodel dendritic solidification, while the material pointmethod [6] was used to compute the stresses and elasto-viscoplastic deformation of the solid. The flow of the liquidwas not simulated and the solidliquid interface was

    1359-6454/$36.00 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

    Corresponding author. Tel.: +1 319 335 5681; fax: +1 319 335 5669.E-mail address: (C. Beckermann).

    assumed to be stress free. In the material point method,Lagrangian point masses are moved through a fixed Eule-rian background mesh. Hence, the material point method iswell suited for simulating large deformations and also forcoupling with the Eulerian phase-field method. However,the issue of contact and bridging between different portionsof a deformed dendrite was not addressed in Ref. [3]. Suchimpingement can lead to the formation of grain bound-aries, even for a single crystal. The formation of grainboundaries between two or more crystals having differentcrystallographic orientations was not treated.

    In the present paper, the model of Ref. [3] is extended toconsider polycrystalline structures. Grain boundaries playan important role in the deformation of a mush, especiallyat high volume fractions of solid. For example, they candelay the formation of solid bridges between dendrites.Not surprisingly, hot tears due to tensile strains in a mushyzone usually form at grain boundaries [7,8]. The inelasticdeformation of multi-grain and dendritic microstructures

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  • M. Yamaguchi, C. Beckermann / Acta Materialia 61 (2013) 22682280 2269

    has been simulated in a few recent studies [9,10], but thosestudies did not consider solidification and the dynamics ofgrain boundaries. Sistaninia et al. [11] developed a three-dimensional (3-D) granular model to study the mechanicalbehavior of a semi-solid mush at high fractions of solid.Although solidification of the initial grain structure wassimulated in Ref. [11], the subsequent stress analysis wasuncoupled. Clearly, the microstructure of the solid playsa key role in the mechanical behavior of a mush. But soliddeformations can also affect the evolution of the solid mor-phology by solidification and grain boundary dynamics.For example, a new grain boundary can form when aseverely deformed dendrite arm grows into an undeformedportion of the same dendrite. Furthermore, new tilt grainboundaries can form when a dendrite arm is bent.

    The grain boundaries are simulated in the present studyusing the polycrystalline phase-field model of Warren et al.[12]. As in all phase-field models, the phase-field parameter/ is used to indicate the local crystalline order, with /= 1 inside the bulk solid and liquid phases, respectively.The solidliquid interface is treated as a diffuse layer ofsmall but finite thickness over which the phase field variessmoothly between / = 1. The grain boundary betweentwo solid grains is also treated as a diffuse interface. Sincethe crystalline order inside a grain boundary is reduced, thephase field assumes values below unity (solid) within thegrain boundary. An additional order parameter, the crystalorientation angle field a, is introduced to measure the localcrystallographic orientation of the solid with respect to afixed coordinate system. If two neighboring grains are mis-oriented, the orientation angle varies smoothly across thediffuse grain boundary from the value in one grain to thevalue in the other grain. The misorientation, Da, is givenby the integral of the orientation angle gradient, $a, acrossthe grain boundary. The phase field and the orientationangle are closely coupled inside a grain boundary. The lar-ger the angle gradient (or misorientation), the lower theminimum value of the phase field. At some critical misori-entation, the minimum value of the phase field reaches /= 1 and the grain boundary is fully wet. The model ofWarren et al. [12] also considers the anisotropy in the inter-facial energy, which is essential for modeling dendriticsolidification. They demonstrated that the model correctlypredicts phenomena such as triple junction behavior, thewetting condition for a grain boundary, curvature-drivengrain boundary motion and grain rotation.

    In the present paper, the polycrystalline phase-fieldmodel of Warren et al. [12] is modified to account fordeformation of the solid. Several numerical examples arepresented to show that the model is correctly implemented.The reader is referred to the companion paper [3] for adescription and detailed validation tests of the materialpoint method for the stress and deformation calculations.A highly simplified description is used for the mechanicalbehavior of a grain boundary. A solid bridge betweentwo adjoining crystals is assumed to be formed when /> 0 inside a grain boundary. Conversely, for values of /

    < 0, the grain boundary is assumed to contain sufficientliquid-like material that it can be considered wet and nostresses are transmitted between the two crystals. The abil-ity of the present model to simulate deformation of poly-crystalline semi-solid structures is demonstrated in severalnumerical examples.

    2. Polycrystalline phase-field method for dendritic

    solidification with solid deformation

    The polycrystalline phase-field model for solidificationof Warren et al. [12] is extended here to include a deforma-tion velocity field, v. It is also modified to reduce exactly tothe quantitative phase-field model of Karma and Rappel[13] for a single dendrite, since that version was used inthe first part of the present study [3].

    Let / denote the phase field, where / = 1 refers to thebulk solid and liquid phases, respectively. The anisotropicform of the two-dimensional (2-D) polycrystalline phase-field evolution equation is given by [12]

    s/w a@/@t v r/

    r W 2w ar/ @f/; kh


    @@xjr/j2Ww a @Ww a


    @@yjr/j2W w a @Ww a


    " #


    sjraj @h/@/




    The above equation is similar to the phase-field equationused in the companion paper [3] for a single dendrite,except for the addition of the last two terms on the right-hand side. These terms account for the effect of crystal ori-entation angle gradients, |$a|, on the phase field. In thepresence of solid deformation, such gradients exist not onlyinside grain boundaries but also inside grains. Inside thelast two terms in Eq. (1), g(/) = h(/) = [(1 + /)/2]2 aremonotonically increasing functions and s and e are anglegradient coefficients that can be related to grain boundaryproperties (see below) [12]. The above phase-field equationalso includes anisotropy in the surface energy of a crystal.Following the methodology of Warren et al. [12], the aniso-tropic phase-field relaxation time and diffuse interfacethickness parameter are given by s/ (w a) = s0n2(w a)and W(w a) = W0n(w a), respectively, where the four-fold anisotropy function is given by n(w a) =1 + ecos [4(w a)] and e is the anisotropy strength. Theinclination [10] angle of