A Novel Physics-Based and Data-Supported Microstructure Model for Part-Scale Simulation of Laser Powder Bed Fusion of Ti-6Al-4V
Jonas Nitzler, Christoph A. Meier, Kei W. Müller, Wolfgang A. Wall, Neil E. Hodge
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OVEL P HYSICS -B ASED AND D ATA -S UPPORTED M ICROSTRUCTURE M ODEL FOR P ART -S CALE S IMULATION OF T I -6A L -4V S ELECTIVE L ASER M ELTING
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Jonas Nitzler*
Institute for Computational MechanicsTechnical University of MunichD-85748 Garching b. München [email protected]
Christoph A. Meier*
Institute for Computational MechanicsTechnical University of MunichD-85748 Garching b. München [email protected]
Kei W. Müller
Institute for Computational MechanicsTechnical University of MunichD-85748 Garching b. München [email protected]
Wolfgang A. Wall
Institute for Computational MechanicsTechnical University of MunichD-85748 Garching b. München [email protected]
Neil E. Hodge
Lawrence Livermore National LaboratoryLivermore, CA 94550-9234 [email protected]
January 15, 2021*shared first authorship A BSTRACT
The elasto-plastic material behavior, material strength and failure modes of metals fabricated byadditive manufacturing technologies are significantly determined by the underlying process-specificmicrostructure evolution. In this work a novel physics-based and data-supported phenomenologicalmicrostructure model for Ti-6Al-4V is proposed that is suitable for the part-scale simulation ofselective laser melting processes. The model predicts spatially homogenized phase fractions of themost relevant microstructural species, namely the stable β -phase, the stable α s -phase as well asthe metastable Martensite α m -phase, in a physically consistent manner. In particular, the modeledmicrostructure evolution, in form of diffusion-based and non-diffusional transformations, is a pureconsequence of energy and mobility competitions among the different specifies, without the need forheuristic transformation criteria as often applied in existing models. The mathematically consistentformulation of the evolution equations in rate form renders the model suitable for the practicallyrelevant scenario of temperature- or time-dependent diffusion coefficients, arbitrary temperatureprofiles, and multiple coexisting phases. Due to its physically motivated foundation, the proposedmodel requires only a minimal number of free parameters, which are determined in an inverseidentification process considering a broad experimental data basis in form of time-temperaturetransformation diagrams. Subsequently, the predictive ability of the model is demonstrated by meansof continuous cooling transformation diagrams, showing that experimentally observed characteristicssuch as critical cooling rates emerge naturally from the proposed microstructure model, instead ofbeing enforced as heuristic transformation criteria. Eventually, the proposed model is exploitedto predict the microstructure evolution for a realistic selective laser melting application scenario a r X i v : . [ c s . C E ] J a n PREPRINT - J
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15, 2021 and for the cooling/quenching process of a Ti-6Al-4V cube of practically relevant size. Numericalresults confirm experimental observations that Martensite is the dominating microstructure species inregimes of high cooling rates, e.g. due to highly localized heat sources or in near-surface domains,while a proper manipulation of the temperature field, e.g., by preheating the base-plate in selectivelaser melting, can suppress the formation of this metastable phase. K eywords Ti-6Al-4V microstructure model · metal additive manufacturing · selective laser melting · part-scalesimulations · inverse parameter identification Additive manufacturing has become an enabler for next-generation mechanical designs with applications ranging fromcomplex geometries for patient-specific implants to custom lightweight structures for the aerospace industry. Especiallymetal selective laser melting (SLM) has gained broad interest due to its high quality and flexibility in the manufacturingprocess of load-bearing structures. Still, the reliable certification of such parts is an open research field not least becauseof a multitude of complex phenomena requiring the modeling of interactions between several physical domains onmacro-, meso- and micro-scale [1].A significant impact on elastoplastic material behavior, failure modes and material strength is imposed by the evolvingmicrostructural composition during the SLM process [2–5]. Modeling the microstructure evolution in selective lasermelting is thus an important aspect for more reliable and accurate process simulations and a crucial step towardscertifiable computer-based analysis for SLM parts.In [3, 6] microstructure models are divided into the categories statistical [7–17], phenomenological [2–4, 6, 18–27]and phase-field [28–32] models. Statistical models are either data-driven and infer statements of coarse-grainedtrends from experiments or apply local stochastic transformation rules and neighborhood dependencies, which mightbe based on physical principles. This category includes in the context of this work also data-based surrogates andmachine learning approaches as well as Monte-Carlo simulations and (stochastic) cellular automaton approaches.Without physical foundation, the reliability and predictive ability of purely data-driven approaches is rather limited,especially in the case of very scarce and expensive experimental data (e.g., dynamic microstructure characteristics in thehigh-temperature regime) or if the available data does not contain certain physical phenomena at all, which might resultin high generalization errors. In case the simulation is based on stochastic rules (e.g., Monte-Carlo simulations) oneencounters often challenges in terms of computationally demands as a reliable response statistic requires a large numberof simulation runs. Furthermore, a consistent conservation of global and local physical properties for the individualsimulation runs remains an open challenge. Stochastic properties might additionally be space and time dependent orfunctionally dependent on further physical properties. The inference of suitable and generalizable parameterizations ofthese stochastic properties is especially problematic in the case of limited experimental data.On the other end of the spectrum of available models, phase-field approaches offer the greatest insight into microstucturalevolution and provide a detailed resolution of the underlying physics-based phenomena of crystal formation anddissolution, such as crystal boundaries and lamellae orientation. However, the resolution of length scales below thesize of single crystals comes at a considerable computational cost, which hampers their application for part-scalesimulations.A preferable cost-benefit ratio can be found in the category of phenomenological modeling approaches. Here, mi-crostructure evolution is described in a spatially homogenized (macroscale) continuum sense by physically motivated,phenomenological phase fraction evolution laws that can be solved at negligible extra cost as compared to standard ther-mal (or thermo-mechanical) process simulations. In this work we will propose a novel physics-based and data-supportedphenomenological microstructure model for Ti-6Al-4V that is suitable for the part-scale simulation of selective lasermelting processes. We present several original contributions, compared to existing approaches of this type. Compared toexisting approaches of this type several original contributions, both in terms of physical and mathematical consistencybut also in terms of the underlying data basis, can be identified.From a physical point of view, the phase fraction evolution equations proposed in this work are solely motivated byenergy considerations, i.e. deviations from thermodynamic equilibrium configurations are considered as driving forcesfor diffusion-based and non-diffusional transformations. Thus, the evolution of the most relevant microstructural phases,namely the β -phase, the stable α s -phase as well as the metastable martensitic α m -phase, is purely driven by an energeticcompetition and the temperature-dependent mobility of these different specifies. This is in strong contrast to existingapproaches, where e.g. the formation of meta-stable phases is triggered by heuristic rules for critical cooling rates,which are taken from experimental observations and explicitly prescribed in the model to match the former. In thepresent approach, however, there is no need to prescribe such critical cooling rates as criterion for phase formation. PREPRINT - J
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Instead, phase formation is a pure consequence of the underlying energy and mobility competition. Critical cooling ratescan be predicted as a result of the modeling approach, and show very good agreement to experimental observations.From a mathematical point of view, the diffusion-based transformations are described in a consistent manner by evolutionequations in differential form, i.e. ordinary differential equations that are numerically integrated in time, which rendersthe model suitable for the practically relevant case of solid state transformations involving temperature or time-dependentdiffusion coefficients, arbitrary temperature profiles, and multiple coexisting phases. Again, this is in contrast to existingapproaches modeling the phase evolution with Johnson-Mehl-Avrami-Kolmogorov (JMAK) equations. In fact, JMAKequations can be identified as analytic solutions for differential equations of the aforementioned type, which are,however, not valid anymore in the considered case of non-constant (temperature-dependent) parameters.From a data science point of view, unknown parameters in existing modeling approaches are typically calibrated onthe basis of single experiment data. As a consequence, this single experiment can then be represented with very goodagreement while an extrapolation of the calibration data, i.e. a truly predictive ability, is only possible within verynarrow bounds. In the present approach, a broad basis of experimental data in form of time-temperature transformation(TTT) diagrams is considered for inverse parameter identification of the (small number of) unknown model parameters.Moreover, the predictive ability of the identified microstructure model is verified on an independent data set in form ofcontinuous-cooling transformation (CCT) diagrams, showing very good agreement in the characteristics (e.g. criticalcooling rates) of numerically predicted and experimentally measured data sets. Since experimental CCT data is verylimited (to only a few discrete cooling curves), the prediction of these diagrams by numerical simulation is not onlyrelevant for model verification. In fact, the proposed microstructure model allowed for the first time to predict CCT dataof Ti-6Al-4V for such a broad and highly resolved range of cooling rates, thus providing an important data basis forother researchers in this field. Eventually, the proposed model is exploited to predict the microstructure evolution for arealistic SLM application scenario (employing a state-of-the-art macroscale SLM model) and for the cooling/quenchingprocess of a Ti-6Al-4V cube with practically relevant size (side length cm ).The structure of the paper is as follows: Section 2 briefly presents the relevant basics of Ti-6Al-4V crystallography, thebasic assumptions and derivation of the proposed microstructure evolution laws and finally the temporal discretizationand implementation of the model in form of a specific numerical algorithm. Section 3 depicts the data-supported inverseparameter identification on the basis of TTT diagrams and model verification in form of CCT diagrams. In Section 4,first the basics of a thermo-mechanical finite element model employed for the subsequent part-scale simulations arepresented. Then, in Sections 4 and 5 applicability of the proposed microstructure model to part scale simulationsis demonstrated by means of two practically relevant examples, a realistic SLM application scenario as well as thecooling/quenching process of a Ti-6Al-4V cube. In the following, we will derive a model for the microstructure evolution in Ti-6Al-4V in terms of volume-averagedphase fractions. First, we introduce fundamental concepts and outline our basic assumptions in Section 2.1. Afterwards,equilibrium and pseudo-equilibrium compositions of the microstructure states are presented in Section 2.2 as a basisfor the subsequent concepts for arbitrary microstructure changes in Section 2.3. The model will be presented in acontinuous and discretized formulation. The latter is then used in the numerical demonstrations.
The aspects of microstructure evolution in Ti-6Al-4V alloys as considered in this work are assumed to be determinedby the current microstructural state, the current temperature T as well as the temperature rate ˙ T , i.e. its temporalderivative. An overview of characteristic temperatures are given in Table 1. When cooling down the alloy from theTable 1: Characteristic temperatures deployed in the microstructure model T α m ,end [K] T α m , sta [K] T α s , end [K] T α s , sta [K] T sol [K] T liq [K]293 848 935 1273 1878 1928- [4, 6, 33] - [4, 22] [34] [34]molten state, solidification takes place between liquidus temperature T liq and solidus temperature T sol . The co-existentliquid and solid phase fractions in this temperature interval shall be denoted as X liq and X sol = 1 − X liq . Below the PREPRINT - J
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15, 2021 solidus temperature T sol the microstructure of Ti-6Al-4V is characterized by body-centered-cubic (bcc) β -crystals andhexagonal-closed-packed (hcp) α -crystals. First, β -crystals will grow in direction of the maximum temperature gradientfor T sol < T ≤ T liq [4]. Depending on the prevalent cooling conditions, the α -phase can be further subdivided intostable α s and metastable α m -phases (Martensite). For sufficiently slow cooling rates | ˙ T | (cid:28) | ˙ T α m , min | the microstructureevolution can follow the thermodynamic equilibrium, i.e. stable α s nucleates into prior β -grains. This diffusion-driventransformation between alpha-transus start temperature T α s , sta and alpha-transus end temperature T α s , end results in atemperature-dependent equilibrium composition X eq α ( T ) (see Figure 1, left) characterized by α s - and β -phase,i.e. phase fractions X α s = 0 . and X β = 0 . , for temperatures below T α s , end [4, 20, 28, 35].Under faster cooling conditions, the formation rates of the stable α s -phase, which are thermally activated and limitedby the diffusion-driven nature of this transformation process, cannot follow the equilibrium composition X eq α ( T ) anymore such that β -phase fractions higher than remain below T α s , end . At temperatures below the Martensite-start-temperature T α m , sta , the metastable Martensite-phase α m becomes energetically more favorable than the excess(transformation-suppressed) β -phase fraction beyond . Under such conditions the β -crystals collapse almost instan-taneously into metastable Martensite following a temperature-dependent (pseudo-) equilibrium composition X eq α m ( T ) [4,6, 22, 33, 36]. The critical cooling rate ˙ T α m , min is defined as the cooling rate above which the formation of stable α s iscompletely suppressed (up to the precision of measurements). The resulting microstructure consists exclusively of β -and α m -phase fractions. The temperature-dependent Martensite phase fraction is denoted as X eq α m , ( T ) for this extremecase. It is typically reported that the Martensite-end-temperature T α m , end , i.e. the temperature when the Martensiteformation is finished, is reached at room temperature T ∞ going along with a maximal Martensite phase fraction of X α m = 0 . in this extreme case (see Figure 1, right).Figure 1: Left: Equilibrium compositions X α s , X β and X liq resulting from slow cooling rates | ˙ T | (cid:28) | ˙ T α m , min | , suchthat X α m = 0 . The experimental data is based on Katzarov [28], Kelly [4], Malinov [20] and Pederson [35]. Right:Metastable (pseudo-)equilibrium composition due to Martensite transformation for | ˙ T | > | ˙ T α m , min | , such that X α s = 0 .In addition to these most essential microstructure features, which will be considered in the present work, some additionalmicrostructural distinctions are often made in the literature [4, 6, 37]. For example, the stable alpha phase α s can befurther classified into its morphologies grain-boundary- α gb and Widmanstätten- α w phase, with X α s = X α gb + X α w .During cooling, grain-boundary- α gb forms first between the individual β -crystals. Upon passing the intergranularnucleation temperature T ig , lamellae-shaped Widmanstätten- α w grow into the prior β -crystal starting from the grainboundary. The appearance of such α -morphologies can have different manifestations such as colony- and basketweave- α s or equiaxed- α s -grains [4].The microstructure model proposed in this work will only consider the accumulated α s phase without further distinguish-ing α gb and α w morphologies. Given their similar mechanical properties [38, 39], this approach seems to be justified,as the proposed microstructure model shall ultimately be employed to inform homogenized, macroscale constitutivelaws for the part-scale simulation of metal powder bed fusion additive manufacturing (PBFAM) processes. In a similarfashion, the so-called massive transformation is often observed in the range of cooling rates that are sufficiently highbut still below ˙ T α m , min , which leads to microstructural properties laying between those of pure α s and pure α m . Forsimilar reasons as argued above, this microstructural species is not explicitly resolved by a separate phase variable inthe present work. Instead, the effective mechanical properties of this intermediate phases are captured implicitly bythe co-existence of α s and α m phase resulting from the present model at these cooling conditions. In addition and insummary, the following basic assumptions are made for the proposed modeling approach: PREPRINT - J
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1. The microstructure is described in terms of (volume-averaged) phase fractions, i.e. no explicit resolution ofgrains and grain boundaries.2. Only the most important phase species β , α s and α m are considered.3. The Martensite-start- and Martensite-end-temperature are considered as constant, i.e. independent of thecurrent microstructure configuration.4. Currently, no information about (volume-averaged) grain sizes, morphologies and orientations is provided bythe model.5. The influence of the mechanical stress state, microstructural imperfections (e.g. dislocations) as well as furthermorphologies is not considered.Partly, these assumptions are motivated by a lack of corresponding experimental data. In our ongoing research work,we intend to address several of these limitations. Remark (Cooling rates during quenching) . Note that the cooling rates during quenching experiments are nottemporally constant in general. Thus, the critical cooling rate ˙ T α m , min measured in experiments is usually thecooling rate at one defined point in time, typically defined at a high temperature value such that the measuredcooling rate is (close to) the maximal cooling rate reached during the quenching process. In such a manner, avalue of ˙ T α m , min = 410 K/s has been reported in the literature for Ti-6Al-4V [33] but was interpreted in severalcontributions [6, 22, 40] as a fixed constraint for Martensite transformation.
In this Section the temperature-dependent, thermodynamic equilibrium and pseudo-equilibrium compositions of the β , α s and α m are described in a quantitative manner. These phase fractions X i ∈ [0; 1] have to fulfill the followingcontinuity constraints: X sol + X liq = 1 , (1a) X α + X β = X sol , (1b) X α s + X α m = X α , (1c)For simplicity, the solidification process between liquidus temperature T liq and solidus temperature T sol is modeled viaa linear temperature-dependence of the solid phase fraction X sol : X sol = for T ≤ T sol , − T liq − T sol · ( T − T sol ) for T sol < T < T liq , for T ≥ T liq . (2)Moreover, we follow the standard approach to model the temperature-dependent stable equilibrium phase frac-tion X eq α ( T ) , towards which the α s -phase tends in the extreme case of very slow cooling rates | ˙ T | (cid:28) | ˙ T α m , min | , on thebasis of an exponential Koistinen-Marburger law ( [41]; see black solid line in Figure 1): X eq α ( T ) = . for T < T α s , end , − exp (cid:2) − k eq α · ( T α s , sta − T ) (cid:3) for T α s , end ≤ T ≤ T α s , sta , for T > T α s , sta . (3)While the alpha-transus start temperature T α s , sta = 1273 K has been taken from the literature [4, 22], the parameters T α s , end = 935 K and k eq α = 0 . K − in (3) have been determined via least-square fitting based on differentexperimental measurements as illustrated in Figure 1 on the left side. It has to be noted that the equilibrium composition X eq α = f ( T ) in form of a temperature-dependent function f ( T ) as given in (3) could alternatively be derived as thestationary point ( ∂ Π(X α s , T ) /∂ X α s ) | (X α s =X eq α ) ˙= 0 ⇔ X eq α − f ( T ) ˙= 0 of a generalized thermodynamic potential Π(X α s , T ) containing contributions, e.g., from the Gibb’s free energies of the individual phases β and α s , fromphase/grain boundary interface energies or from (transformation-induced) strain energies [2]. Here, the dependence ofthe potential Π(X α s , T ) on X β has been omitted since X β = 1 − X α s for solid material under equilibrium conditions,i.e. in the absence of Martensite. In the present work, for simplicity, the expression for the equilibrium composition X eq α = f ( T ) has directly been postulated and calibrated on experimental data instead of formulating the individualcontributions of a potential, which would involve additional unknown parameters. Still, from a mathematical point ofview it shall be noted that the expression X eq α = f ( T ) in (3) is integrable, i.e., a corresponding potential can be found ingeneral, resulting in beneficial properties not only of the physical model but also of the numerical formulation. PREPRINT - J
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Next, we consider the second extreme case of very fast cooling rates | ˙ T | ≥ | ˙ T α m , min | at which the diffusion-drivenformation of X α s is completely suppressed. For this case, we model the metastable Martensite pseudo equilibriumfraction X eq α m , ( T ) , emerging in the absence of α s -phase, based on an exponential law [4, 6, 22, 41, 42]: X eq α m , ( T ) = . for T < T ∞ , − exp (cid:2) − k eq α m ( T α m , sta − T ) (cid:3) for T ∞ ≤ T ≤ T α m , sta , for T > T α m , sta . (4)While the value T α m , sta = 848 K has been taken from the literature [4, 6, 33], we choose k eq α m = 0 . K − suchthat (4) yields a maximal Martensite fraction of X eq α m ( T ∞ ) = 0 . at room temperature, which is in agreement tocorresponding experimental observations [22] (see Figure 1 on the right).Finally, we want to consider the most general case of cooling rates that are too fast to complete the diffusion-drivenformation of the stable α s phase before reaching the Martensite start temperature T α m , sta but still below the critical rate | ˙ T α m , min | , i.e. a certain amount of stable α s phase has still been formed and consequently a Martensite phase fractionbelow is expected at room temperature. For this case, we postulate an effective pseudo equilibrium phase fraction X eq α m ( T ) for the α m phase that accounts for the reduced amount of transformable β -phase at presence of a given phasefraction X α s of the stable α s phase according to X eq α m ( T ) = X eq α m , ( T ) · (0 . − X α s )0 . . (5)It can easily be verified that (5) fulfills the important relation X eq α m ( T )+X α s < . for arbitrary values of the currenttemperature T and α s -phase fraction X α s . This means, for any given value X α s , an instantaneous Martensiteformation according to X eq α m ( T ) will never result in a total α -phase fraction X α = X α m + X α s that exceeds thecorresponding equilibrium composition X eq α (which takes on a value of X eq α = 0 . in the relevant temperaturerange below T α s , end ). In the extreme case that the maximal α s -phase fraction of has already been formedbefore reaching the Martensite start temperature T α m , sta , Equation (5) ensures that no additional Martensiteis created during the ongoing cooling process. Again, the pseudo equilibrium composition X eq α m = ˜ f ( T ) inform of a temperature-dependent function ˜ f ( T ) as given in (5) could alternatively be derived as the stationarypoint ( ∂ Π(X α m , X α s , T ) /∂ X α m ) | (X α m =X eq α m ) ˙= 0 ⇔ X eq α m − ˜ f ( T ) ˙=0 of a generalized thermodynamic potential Π(X α m , X α s , T ) , in which the current α s phase fraction X α s (cid:54) = X eq α can be considered as a fixed parameter. Thus, X eq α m = ˜ f ( T ) according to (5) is not a global minimum of this generalized potential but rather a local minimum withrespect to X α m under the constraint of a given α s phase fraction X α s (cid:54) = X eq α . This model seems to be justified given theconsiderably slower formation rate of the α s phase as compared to the (almost) instantaneous Martensite formation (seealso the next section).For a given temperature T α m , sta ≥ T ≥ T ∞ and α s -phase fraction X α s ≤ . during a cooling experiment, Equation (5)will in general yield a Martensite phase fraction such that X α = X α s + X α m ≤ . , i.e. the sum of stable and martensiticalpha phase fraction might be smaller than the equilibrium phase fraction X eq α according to (3). In this case of co-existing α s - and α m -phase, it is assumed that X eq α in (3) as well as its complement X eq β = 1 − X eq α represent the (pseudo-)equilibrium compositions for the total α phase fraction X α = X α s + X α m and for the β phase fraction X β = 1 − X α .In other words, the driving force for diffusion-based α s -formation, as discussed in the next section, is assumed to resultin the following long-term behavior: lim t →∞ X α = X eq α ⇔ lim t →∞ X β = X eq β with X α = X α s + X α m , X β = 1 − X α , X eq β = 1 − X eq α . (6)While at low temperatures, Martensite is energetically more favorable than the β -phase, which is the driving forcefor the instantaneous Martensite formation, it is assumed to be less favorable than the α s -phase. Therefore, thereexists a driving force for a diffusion-based dissolution of Martensite into α s -phase, resulting in the following long-termbehavior: lim t →∞ X α m = ¯X eq α m = 0 . (7)However, as discussed in the next section, the diffusion rates for this thermally activated process drop to (almost) zeroat low temperatures such that Martensite is retained as meta-stable phase at room temperature. Thus, (7) can only beconsidered as theoretical limiting case in this low temperature region. PREPRINT - J
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Since the melting and solidification process is completely described by Equation (2), this section focuses on solid-state phase transformations for temperatures
T < T sol below the solidus temperature (i.e. X sol = 1 ). To model theformation and dissolution of the α s -, α m - and β -phase, we propose evolution equations in rate form with the followingcontributions to the total rates, i.e. to the total time derivatives ˙X α s , ˙X α m and ˙X β , of the three phases: ˙X α s = ˙X β → α s + ˙X α m → α s − ˙X α s → β , (8a) ˙X α m = ˙X β → α m − ˙X α m → α s − ˙X α m → β , (8b) ˙X β = ˙X α s → β + ˙X α m → β − ˙X β → α s − ˙X β → α m . (8c)Here, e.g., ˙X β → α s represents the formation rate of α s out of β while ˙X α s → β represents the dissolution rate of α s to β .The meaning of the individual contributions in (8), the underlying transformation mechanisms (e.g. instantaneous vs.diffusion-based) as well as the proposed evolution laws will be discussed in the following. Since the formation anddissolution of phases might follow different physical mechanisms in general, we have intentionally distinguished the(positive) rates ˙X x → y ≥ and ˙X y → x ≥ of two arbitrary phases x and y instead of describing both processes viapositive and negative values of one shared variable ˙X y ↔ x . It is obvious that (8) satisfies the continuity Equation (1) fortemperatures T < T sol below the solidus temperature (with X sol = 1 and ˙X sol = 0 ), which reads in differential form: ˙X α s + ˙X α m + ˙X β = 0 if X sol = 1 . (9a)Thus, the phase fraction X β = 1 − X α s − X α m ∀ T < T sol can be directly calculated from (1) and only the evolutionequations for the phases α s and α m will be considered in the numerical algorithm presented in Section 2.3.2. In the following, the individual contributions to the transformation rates in (8) will be discussed. One of the mainassumptions for the following considerations is that α m ↔ β transformations take place on much shorter time scalesthan α s ↔ β transformations [4, 5, 21], which allows to consider the former as instantaneous processes while the latterare modeled as (time-delayed) diffusion processes. In a first step, the diffusion-based formation of the stable α s -phaseout of the β -phase is considered [34, 37]. Modified logistic differential Equations [43] can be considered as suitablemodel and powerful mathematical tool to model diffusion processes of this type. Based on this methodology, wepropose the following model for the diffusion-based transformation β → α s : ˙X β → α s = k α s ( T ) · (X α s ) cα s − cα s · (cid:16) X β − X eq β (cid:17) cα s +1 cα s for X β > X eq β , else . (10)Diffusion equations of this type typically consist of three factors: i) The factor (X β − X eq β ) represents the driving forceof the diffusion process in terms of transformable β phase (see (6)) and has a decelerating effect on the transformationduring the ongoing diffusion process. With the continuity relations X β = 1 − X α and X eq β = 1 − X eq α this term couldalternatively be written as (X eq α − X α s ) . ii) The factor with X α s leads to a transformation rate that increases withincreasing amount of created α s -phase, i.e., it has an accelerating effect on the transformation rate during the ongoingdiffusion process. Physically, this term can be interpreted as representation of the diffusion interface between α s - and β -phase, which increases with increasing size of the α s -nuclei (and thus with increasing α s -phase fraction). iii) Thefactor k α s ( T ) represents the temperature-dependent diffusion rate of this thermally activated process. From a physicalpoint of view, this term considers the temperature-dependent mobility of the diffusing species. When plotting the phasefraction X α s over time (at constant temperature), the factors i) and ii) together result in the characteristic S-shape ofsuch diffusion-based processes as exemplary depicted in Figure 2 for four combinations of k and c . Depending on thetype of diffusion process, different values for the exponent c α s can be derived from the underlying physical mechanismsresulting in more process-specific types of diffusion equations. In its most general form, which is applied in this work,the exponent c α s of the modified logistic differential equation is kept as a free parameter, which allows an optimalinverse identification based on experimental data even for complex diffusion processes [43].When cooling down ( ˙ T < ) the material at temperatures below the Martensite start temperature ( T < T α m , sta ) and theequilibrium composition of the stable α s -phase has not been reached yet ( X α s < X eq α ), an instantaneous Martensiteformation out of the (excessive) β -phase is assumed following the Martensite pseudo equilibrium composition X eq α m according to (5). Mathematically, this modeling assumption can be expressed by an inequality constraint based on thefollowing Karush-Kuhn-Tucker (KKT) conditions: X α m − X eq α m ≥ ∧ ˙X β → α m ≥ ∧ (X α m − X eq α m ) · ˙X β → α m = 0 . (11) PREPRINT - J
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15, 2021 X [ − ] k = 0.1, c = 2.0k = 0.3, c = 2.0k = 0.1, c = 4.0k = 0.3, c = 4.0 Figure 2: Four exemplary phase evolutions of a nucleation process modelled with Equation (10) with initial phasefraction X = 0 , equilibrium X eq = 1 and combinations of k ∈ { . , . } and c ∈ { . , . } .The constraint X α m − X eq α m ≥ (first inequality in (11)) states that X α m cannot fall below the equilibrium composition X eq α m since Martensite will be formed instantaneously out of the β -phase. Here, the formation rate ˙X β → α m (secondinequality in (11)) plays the role of a Lagrange multiplier enforcing the constraint X α m − X eq α m = 0 as long as Martensiteis formed. According to the complementary condition (third equation in (11)), this formation rate vanishes in caseof excessive X α m -phase (i.e. ˙X β → α m = 0 if X α m − X eq α m > ). This scenario can occur e.g. during heating of theMartensite material (i.e. ˙X eq α m < ) since the diffusion-based Martensite-dissolution process (see below) cannot followthe decreasing equilibrium composition X eq α m in an instantaneous manner.Since the α m -phase is energetically less favorable than the α s -phase, there is a driving force for the transformation α m → α s . In contrast to the instantaneous formation of Martensite, the α m -dissolution to α s is modeled as (time-delayed)diffusion process [6] according to ˙X α m → α s = (cid:40) k α s ( T ) · (X α s ) cα s − cα s · (cid:0) X α m − ¯X eq α m (cid:1) cα s +1 cα s for X α m > ¯X eq α m , else , (12)where ¯X eq α m = 0 represents the long-term equilibrium state ( t → ∞ ) of the Martensite phase (see Equation (7)).Equations (10) and (12) have the same structure and share the same exponent c α s as well as the same diffusion rate k α s ( T ) since both describe the diffusion-based formation of α s -phase, i.e., both result in the same daughter phase. Theunderlying modeling assumption is that the transformation α m → α s can be split according to α m → β → α s , i.e., it isassumed that Martensite first dissolves instantaneously into the intermediate phase β , which afterwards transforms tothe α s -phase in a diffusion-based β → α s process similar to (10). The model for the temperature-dependence of k α s ( T ) as described below will result in diffusion rates that drop to (almost) zero at low temperatures such that Martensite isretained as meta-stable phase at room temperature, which is in accordance to the corresponding experimental data.Eventually, also the case of lacking β -phase (i.e., X β < X eq β ) shall be considered: Due to the nature of the β -dissolutionprocesses ˙X β → α s according to (10) and ˙X β → α m according to (11), which only yield contributions as long as X β > X eq β ,this scenario can only arise from ˙X eq β > , i.e., if ˙ T > and T ∈ [ T α s , end ; T α s , sta ] .Experimental data by Elmer et al. [34] strongly supports a diffusional behavior of β -phase build up, respectively α s -dissolution during heating of Ti-6Al-4V. We thus chose a diffusional nucleation model in Equation (13) for the PREPRINT - J
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15, 2021 resulting α s → β transformation: ˙X α s → β = k β ( T ) · (cid:16) ˜X β (cid:17) cβ − cβ · (cid:0) X α − X eq α (cid:1) cβ +1 cβ , for X α > X eq α , else. (13)Here, ˜X β = X β − . defines a corrected β -phase fraction. This ensures that the second factor in (13), which can beinterpreted as a measure for the increasing diffusion interface during the formation process, starts at a value of zerowhen heating material with initial equilibrium composition X β = X eq β above T α s , end . By this means, the temporalevolution of ˜X β (i.e., of the additional β -material beyond ) begins with a horizontal tangent when exceeding T α s , end as also observable in corresponding experiments [34]. From a physical point of view this experimental observation - aswell as the corresponding diffusion model in (13) - rather correspond to a phase nucleation and subsequent growthprocess of a new phase fraction ˜X β (starting at ˜X β = 0 ) than a continued growth process of a pre-existing phasefraction X β (starting at X β = 0 . ). While existing experimental data in terms of temporal phase fraction evolutions isvery limited, there is still significant experimental evidence that the α s → β dissolution should be considered as a(time-delayed) diffusion-based process [4, 34] rather than an instantaneous transformation, which for simplicity hasbeen assumed in some existing modeling approaches, like e.g., [6].Let us again consider the case of lacking β -phase (i.e. X β < X eq β ), which can only occur in the temperature interval T ∈ [ T α s , end ; T α s , sta ] as discussed in the paragraph above. If in this scenario, a certain amount of remaining Martensitematerial (i.e. X α m > ) exists, it is assumed that the α m -phase fraction is decreased in an instantaneous α m → β transformation such that X β can follow the energetically favorable equilibrium composition X eq β . Mathematically, thismodeling assumption can be expressed by an inequality constraint with the following KKT conditions:If X α m > β − X eq β ≥ ∧ ˙X α m → β ≥ ∧ (X β − X eq β ) · ˙X α m → β = 0 . (14)The constraint X β − X eq β ≥ (first inequality in (14)) states that X β cannot fall below the equilibrium composition X eq β as long as a remainder of Martensite, instantaneously transformable into X β , is present. Again, the formation rate ˙X α m → β (second inequality in (14)) plays the role of a Lagrange multiplier enforcing the constraint X β − X eq β = 0 as longas the α m → β transformation takes place. According to the complementary condition (third equation in (14)), thisformation rate vanishes in case of excessive β -phase (i.e. ˙X α m → β = 0 if X β − X eq β > ). Since in the considered scenario,the remaining contributions to ˙X β vanish, i.e., ˙X α s → β = 0 and ˙X β → α s = 0 due to X β = X eq β as well as ˙X β → α m = 0 dueto T > T α m , sta , (8c) together with the differential form of the constraint ˙X β = ˙X eq β allows to explicitly determine thecorresponding Lagrange multiplier to ˙X α m → β = ˙X eq β . Once all Martensite is dissolved, i.e., X α m = 0 , the β -phasefraction cannot follow the corresponding equilibrium composition X eq β in an instantaneous manner anymore, but ratherin a time-delayed manner based on the diffusion process (13).To close the system of model equations proposed in this section, a specific expression for the temperature-dependenceof the diffusion rates k α s ( T ) and k β ( T ) required in (10), (12) and (13) has to be made. The mobility of the diffusingspecies, which is represented by these diffusion rates, is typically assumed to increase with temperature. However, it isassumed that the diffusion rates do not increase in a boundless manner but rather show a saturation at a high temperaturelevel. Moreover, for the considered class of thermally-activated processes, these diffusion rates are assumed to dropto zero at room temperature. The following type of logistic functions represent a mathematical tool for the describedsystem behavior: k α s ( T ) := k − k · ( T − k )] (15)The free parameters k , k , k and c α s governing the β → α s -diffusion processes (10) and (12) will be inverselydetermined in Section 3 based on numerical simulations and experimental data for time-temperature-transformations(TTT). The same temperature-dependent characteristic as in (15) is also assumed for the α s → β -diffusion process (13).Since the dissolution of α s - into β -phase is reported to take place at higher rates [4, 34] as compared to the α s -formationout of β -phase, we allow for an increased diffusion rate k β ( T ) of the form: k β ( T ) := f · k α s ( T ) with f > . (16)Thus, only the two free parameters f and c β are required for the α s → β -diffusion. These two parameters will beinversely determined in Section 3 based on heating experiments taken from [34]. PREPRINT - J
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Remark (Martensite cooling rate) . In our model the critical cooling rate ˙ T α m , min = − K/s is not prescribedas an explicit condition for Martensite formation as done in existing microstructure modeling approaches [6,40]. Instead, the process of Martensite formation is a pure consequence of physically motivated energy balancesand driving forces for diffusion processes. In Section 3.3 it will be demonstrated that a value very close to ˙ T α m , min = − K/s results from the present modeling approach in a very natural manner when identifyingthe critical rate for pure Martensite formation from continuous-cooling-transformation (CCT) diagrams creatednumerically by means of this model.
Remark (Johnson-Mehl-Avrami-Kolmogorov (JMAK) equations) . In other publications [4, 6, 19, 23, 27, 34,40, 44], Johnson-Mehl-Avrami-Kolmogorov (JMAK) equations are used to predict the temporal evolution ofthe considered phase fractions. It has to be noted that JMAK equations are nothing else than analytic solutionsof differential equations (for diffusion processes) very similar to (10), which are, however, only valid in caseof constant parameters k α s , X α s and X eq β . Since these parameters (due to their temperature-dependence) are notconstant for the considered class of melting problems, only a direct solution of the differential equations vianumerical integration, as performed in this work, can be considered as mathematically consistent. We furthermorewant to note that the mathematical form of JMAK equations, which involve logarithmic and exponential expressions,is prone to numerical instabilities in practical scenarios, especially for the extremely high temperature rates thatappear in SLM processes. In contrast, the proposed algorithm in the following Section 2.3.2 has a simple and robustmathematical character. For the numerical solution of the microstructural evolution laws from the previous section, we assume that a temporallydiscretized temperature field ( T n , ˙ T n ) based on a time step size ∆ t is available at each discrete time step n (e.g.provided by a thermal finite element model as presented in Section 4.1). In principle, any time integration scheme canbe employed for temporal discretization of the phase fraction evolution equations from the last section. Specifically, inthe subsequent numerical examples either an implicit Crank-Nicolson scheme or an explicit forward Euler scheme havebeen applied. For simplicity, the general algorithmic realization of the time-discrete microstructure evolution model isdemonstrated on the basis of a forward Euler scheme.For the following time integration procedure of (8) it has to be noted that only the rates ˙X α m → α s , ˙X β → α s , ˙X α s → β corresponding to diffusion processes will be integrated in time. Instead of integrating the rates ˙X β → α m and ˙X α m → β cor-responding to instantaneous Martensite formation and dissolution processes, the associated constraints in (11) and (14)will be considered directly by means of algebraic constraint equations. In a first step, assume that the temperature data T n +1 , ˙ T n +1 of the current time step n + 1 as well as the microstructure data X nα s , X nα m , ˙X nα m → α s , ˙X nβ → α s , ˙X nα s → β of thelast time step n is known. With this data, the microstructure update is performed: X n +1 α s = X nα s + ∆ t ( ˙X nβ → α s + ˙X nα m → α s − ˙X nα s → β ) and X n +1 α m = X nα m − ∆ t ˙ X nα m → α s . (17)The time integration error in (17) might lead to a violation of the scope X n +1 α s , X n +1 α m ∈ [0; 0 . of the phase fractionvariables. In this case the relevant phase fraction variable is simply limited to its corresponding minimal or maximalvalue, respectively. Similarly, if X n +1 α = X n +1 α s + X n +1 α m exceeds the maximum value of . the individual contributions X n +1 α s and X n +1 α m are reduced such that the maximum value X n +1 α = 0 . is met and the ratio X n +1 α s / X n +1 α m is preserved.Subsequently, the β -phase fraction is calculated from the continuity equation X n +1 β = 1 − X n +1 α . Afterwards, theupdated equilibrium phase fractions X eq ,n +1 α s and X eq ,n +1 α m are calculated according to (3)-(5) with X n +1 α s and T n +1 before the corresponding equilibrium composition X eq ,n +1 β = 1 − X eq ,n +1 α s of the β -phase is updated. Next, a potentialinstantaneous Martensite formation out of the β -phase according to (11) is considered as follows:If X n +1 α m < X eq α m ,n +1 , Update: X n +1 β ← X n +1 β + X n +1 α m − X eq α m ,n +1 , (18a)Set: X n +1 α m = X eq α m ,n +1 . (18b)Similarly, a potential instantaneous Martensite dissolution into β -phase according to (14) is considered as follows:If X n +1 β < X eq β ,n +1 ∧ X n +1 α m > , Update: X n +1 α m ← X n +1 α m + X eq β ,n +1 − X n +1 β , (19a)Set: X n +1 β = X eq β ,n +1 . (19b)Again, if necessary the increment in (19a) is limited such that the updated phase fraction X n +1 α m does not becomenegative. As a last step, the diffusion-based transformation rates ˙X n +1 β → α s , ˙X n +1 α m → α s and ˙X n +1 α s → β according to (10), (12) PREPRINT - J
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15, 2021 and (13), all evaluated at time step n + 1 , are calculated. With these results, the next time step n + 2 can be calculatedstarting again with (17). Remark (Initial conditions for explicit time integration) . While the diffusion process according to Equation (12)will start at a configuration with X α m (cid:54) = 0 and X α s (cid:54) = 0 , Equation (10) needs to be evaluated for X α s = 0 to initiatethe diffusion process. However, the evolution of Equation (10) based on an explicit time integration scheme willremain identical to zero for all times for a starting value of X α s = 0 . Therefore, during the first cooling periodthe phase fraction has to be initialized at the first time step t n where X eq α > . In the following, the initializationprocedure considered in this work is briefly presented. In a first step, (10) is reformulated using the relations X β = 1 − X α and X eq β = 1 − X eq α as well as the approximate assumptions k α s ( T ) = const. , X eq α = const. and X α m = 0 (i.e., X α = X α s ) for the initial state [43]: ˙ g = ˜ k · g cα s − cα s · (1 − g ) cα s +1 cα s with g = X α X eq α , ˜ k = k α s · X eq α . (20)Based on the analytic solution in [43], evaluated after one time step ∆ t , the initialization for X nα s at t n reads: X nα s = X eq α · (cid:20) (cid:18) c α s ˜ k ∆ t (cid:19) c α s (cid:21) − . (21)We compared this approach with an implicit Crank-Nicolson time integration (either used for the entire simulationor only for initialization of the first time step), where the initial α s -phase fraction X α s does not need to be setexplicitly and found no differences in the resulting diffusion dynamics according to (10). The four parameters θ diff ,α s = [ c α s , k , k , k ] T according to Equations (10), (12) and (15) as well as the two parameters θ diff ,β = [ c β , f ] T according to Equations (13) and (16), are so far still unknown and need to be inversely identifiedvia experimental data sets. For the inverse identification of θ diff ,α s , we use so-called time-temperature-transformation(TTT) experiments, a well-known experimental characterization procedure for microstructural evolutions [4, 6, 44, 45].As TTT-experiments only capture the dynamics of cooling processes, we will identify the parameters θ diff ,β governingthe heating dynamics of the microstructure via data from heating experiments taken from [34]. α s -formation dynamics via TTT-data Time-temperature transformation (TTT) experiments [46] are one of the most important and established proceduresfor (crystallographic) material characterization. The goal of the TTT investigations is to understand the isothermal transformation dynamics of an alloy by plotting the percentage volume transformation of its crystal phases over time.Thereto, the material is first equilibriated at high temperatures such that only the high-temperature phase is present.Afterwards, the material is rapidly cooled down to a target temperature at which it is then held constant over time so thatthe isothermal phase transformation at this temperature can be recorded. Rapid cooling refers here to a cooling rate thatis fast enough so that diffusion-based transformations during the cooling itself can be neglected and can subsequentlybe studied under isothermal conditions at the chosen target temperature. The procedure is repeated for successivelyreduced target temperatures. The emerging diagram of phase contour-lines over the T × log( t ) space is commonlyreferred to as TTT-diagram.In the present work, the simulation of TTT-curves for Ti-6Al-4V was conducted as follows: We initialized themicrostructure state at T = 1400 K > T α s , end with pure β -phase, such that X β = 1 . . Afterwards, the microstructurewas quickly cooled down with ˙ T = − K/s to a target temperature T target ∈ [350 , and the evolution of themicrostructure at this target temperature was recorded over time. The range of target temperatures was discretized insteps of K such that 95 individual target temperatures and hence microstructure simulations were considered. Figure3 depicts the isolines of simulated X α s - (left) and X α m -phase fractions (right), after identifying the model parametersvia the experimental data [4, 6, 44, 47] shown in the left figure. Please note that the phase-fractions in Figure 3 werenormalized with X eq α ( T ) , which takes on a value of . for temperatures below T α s , end , as this was also the case in theunderlying experimental investigations. We highlighted three temperatures T = { , , } K to discuss themicrostucture evolution at these points. We chose a constant cooling rate here as the detailed cooling dynamics are not important for TTT diagrams as long as coolingtakes place "fast enough". In the latter respect, we also investigated several higher cooling rates without observing differences in theresulting TTT-diagrams. PREPRINT - J
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15, 2021 -3 -2 -1 t [s]40060080010001200 T [ K ] T= 400 KT= 800 KT= 1000 KT (cid:1) m ,sta = 848 KT (cid:1) s ,end = 935 KT (cid:1) s ,sta = 1273 K TTT-diagram for X (cid:0) s . . . . . . Malinov (resistivity, 2001): X (cid:1) s = 0.05Malinov (resistivity, 2001): X (cid:1) s = 0.01Kelly: X (cid:1) s = 0.01Malinov (resistivity, 2001): X (cid:1) s = 0.5Malinov (resistivity, 2001): X (cid:1) s = 0.95Malinov (resistivity, 2001): X (cid:1) s = 0.95 10 -3 -2 -1 t [s] T= 400 KT= 800 KT= 1000 KT (cid:1) m ,sta = 848 KT (cid:1) s ,end = 935 KT (cid:1) s ,sta = 1273 K TTT-diagram for X (cid:2) m . . . . Figure 3: Simulation of the TTT-diagram for the α s - and α m -phases using the maximum likelihood point estimatefor the uncertain kinetic parameters θ ∗ diff ,α s of the microstructure evolution, along with experimental data by Malinov[47] and Kelly [37]. Left: Contour-lines for X α s ; Right: Contour-lines for X α m . Contour lines are shown for the , , , , and normalized phase fractions. Three temperatures are marked in red and discussed in theanalysis.At T = 1000 K (between T α s , sta and T α s , end , i.e. above T α m , sta ) we get a pure β → α s transformation. When lookingat higher target temperatures T > K it can be observed that the isoline with 1% phase fraction for X α s is shiftedto longer times, which results from a decreasing value of X eq α (i.e. a decreasing driving force) that slows down theinitial dynamics of X α s -formation at elevated temperatures, even-though k α s is already saturated at its maximal value inthis temperature range (see Figure 4). At T = 800 K we are now below T α m , sta , so that the initial cool-down results(instantaneously) in a Martensite phase fraction according to X eq α m . With ongoing waiting time, the remaining β -phasetransforms in a diffusion-driven manner into stable α s -phase. In the TTT-diagram, we can already notice that the isolinewith 1% phase fraction for X α s is shifted to longer times as compared to the higher temperature level T = 1000 K ,which, this time, is caused by the decreasing value of k α s for lower temperatures, as depicted in Figure 4 and modeledin Equation (15). Moreover, the right-hand side of Figure 3 shows that the Martensite phase fraction decreases again forwaiting times t > s , which represents the diffusion-based dissolution of Martensite into α s -phase according to (12).Finally, the transformation at T = 400 K initially results in almost the maximal possible amount of Martensite(Remember: The -isoline in Figure 3 corresponds to a phase fraction of . for T < T α s , end ). As the diffusionrate k α s is almost zero for such low temperatures, the Martensite phase cannot be dissolved to stable α s in finite timesand the Martensite phase remains present as metastable phase at low temperatures, which agrees well with experimentalobservations. In the TTT-diagram this effect shifts the isolines asymptotically to t → ∞ when approaching the roomtemperature. All in all, the shift to longer times due to a low value of k α s for low temperatures and the delay effect dueto a decreasing (driving force) value of X eq α at high temperatures leads to the typical C-shape of TTT-phase-isolines.Inverse parameter identification was conducted for the diffusion parameters θ diff ,α s by maximizing the data’s likelihoodfor θ diff ,α s . We assumed a (conditionally independent) static Gaussian noise for the measurements on the log( t ) -scale.This assumption is equivalent to a log-normal distributed noise in the data along the time-scale. The maximum-likelihood point estimate can then be determined by solving the following least-square optimization problem (seeRemark below for more details): θ ∗ diff ,α s = argmin θ diff ,α s (cid:88) i (cid:18) X α s , TTT (log( t i ) , T i , θ diff ,α s ) − X α s , TTT , exp .,i (cid:19) (22)In Equation (22) the term X α s , TTT (log( t i ) , T i , θ diff ,α s ) describes the simulated TTT-phase fractions (normalizedby X eq α ( T ) ) at time t i , Temperature T i and for diffusional parameters θ diff ,α s . The index i marks here the specifictemperatures and times for which the corresponding observed experimental data X α s , TTT , exp .,i was recorded. PREPRINT - J
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We utilized a Levenberg-Marquardt optimization routine [48], which is implemented in our in-house software framework
QUEENS [49] to iteratively solve Equation (22). The result of this optimization procedure is given by the parameterset θ ∗ diff ,α s = [ c α s , k , k , k ] T = [2 . , . , . , . T . Additionally, Figure 4 visualizes the temperature-dependent diffusion rate k α s ( T, θ ∗ diff ,α s ) according to Equation (15) that results from these parameters.
400 600 800 1000 1200 1400 T [ K ]0 . . . . . . . k α s ( T , θ ∗ d i ff , α s ) Figure 4: Temperature-dependent diffusion rate k α s ( T, θ ∗ diff ,α s ) for α s -formation resulting from inverse parameteridentification. Remark (Assumptions for inverse identification) . The least-squares optimization problem in Equation (22) is theresult of the following assumptions for the inverse identification task of θ ∗ diff ,α s : The simulation model is expressedby a mapping y ( θ , c ) , with θ being model parameters that should be inferred from data via inverse analysis and c being coordinates on which the model output is recorded. Further model inputs (e.g., parameters that we want tokeep fixed) are omitted to avoid cluttered notation. We assume that a vector of n obs scalar experimental observations y obs , C , recorded at coordinates C = { c i } , was disturbed by conditionally independent and static Gaussian noisewith variance σ n on the log( t ) -space. This assumption is equivalent to log-normal distributed noise on the t -space.Statistically, the former assumption is expressed by the so called likelihood function l ( θ ) = p ( y obs , C | y ( θ , C )) ,which is the probability density for the observed data y obs , C , given a specific choice of the parameterized simulationmodel y ( θ , C ) , evaluated at the same coordinates C . Gaussian conditional independent noise implies now that(theoretically) repeated observations of experimental data y obs , c i at each location c i will result in n obs Gaussianprobability distributions with variance σ n for y obs , c i . Independence in particular means that a disturbance by noiseat c i does not influence the disturbance by noise at any other point. Ideally, the simulation output should reflectthe mean of the observation noise at each c i . The latter can be expressed by the product (due to the conditionalindependence) of n obs Gaussian distributions with their mean values being the simulation outputs y ( θ , c i ) at each c i and with variance σ n : l ( θ ) = p ( y obs , C | y ( θ , C )) = n obs (cid:81) i =1 N (cid:0) y obs , c i | y ( θ , c i ) , σ n (cid:1) ∝ exp (cid:104) − (cid:80) i ( y obs , c i − y ( θ , c i )) σ n (cid:105) .Despite l ( θ ) being a true probability density function for y obs , C , the latter is not the case w.r.t. the model parameters θ , such that l ( θ ) is mostly referred to as the likelihood function . The maximum likelihood (ML) point estimate θ ∗ refers then to a value of θ that maximizes the likelihood function l ( θ ) , respectively the probability density of theobservations y obs , C for the specific choice θ ∗ . The maximum of l ( θ ) can be found by minimizing the sum of thesquare terms in the argument of the exponential function, which leads ultimately to Equation 22. So far, the inverse identification of θ diff ,α s via TTT-data only accounts for the cooling dynamics of the microstructuremodel. In the following, we inversely identify the parameter set θ diff ,β via experimental data from [34], which consistsof three temperature and β -phase data-sets for three positional measurements for an electron beam welding processwith Ti-6Al-4V. Note that the specific measurement positions x = { . , . , . } mm defined in the original work [34]are not relevant here since we only aim at correlating temperature and phase fraction data. Please note also that thisidentification step reuses θ ∗ diff ,α s , respectively the temperature characteristics of k α s as identified above and only scalesthe latter by the factor f (see Equation (16)). The maximum likelihood estimate θ ∗ diff ,β was again calculated by solving PREPRINT - J
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15, 2021 a least-squares optimization problem with the Levenberg-Marquardt optimizer: θ ∗ diff ,β = argmin θ diff ,β (cid:20) (cid:90) (cid:16) X β ( t, T exp. ,x =4 . ( t ) , θ diff ,β ) − X β, exp .,x =4 . ( t ) (cid:17) dt + (cid:90) (cid:16) X β ( t, T exp. ,x =5 . ( t ) , θ diff ,β ) − X β, exp .,x =5 . ( t ) (cid:17) dt + (cid:90) (cid:16) X β ( t, T exp. ,x =5 . ( t ) , θ diff ,β ) − X β, exp .,x =5 . ( t ) (cid:17) dt (cid:21) (23)Equation (23) accounts for all three phase-temperature measurements at locations = { . , . , . } mm simultaneously leading to a robust point estimate θ ∗ diff ,β with low generalization error. The minuends X β ( t, T exp. ,x =4 . ( t ) , θ diff ,β ) , X β ( t, T exp. ,x =5 . ( t ) , θ diff ,β ) and X β ( t, T exp. ,x =5 . ( t ) , θ diff ,β ) reflect the simulations re-sults for the β -phase fraction over time, for the experimental temperature profiles T exp. ,x =4 . ( t ) , T exp. ,x =5 . ( t ) and T exp. ,x =5 . ( t ) , respectively (see Figure 5 second row). The subtrahends X β, exp .,x =4 . ( t ) , X β, exp .,x =4 . ( t ) and X β, exp .,x =5 . ( t ) are the corresponding experimentally measured β -phase fraction profiles over time.The inverse identification yields the parameter set θ ∗ diff ,β = [ c β , f ] T = [11 . , . T . The comparison of experimentaldata from [34] and the simulation-based prediction in Figure 5 show very good agreement. We summarize that the X (cid:1) [ (cid:2) ] Elmer x= 5.5mmSimulation x= 5.5mm0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t [s]8001000120014001600 T [ K ] Elmer: Temperature profile at x= 5.5mm0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t [s]0.00.20.40.60.81.0 X (cid:1) [ (cid:2) ] Elmer x= 5.0mmSimulation x= 5.0mm0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t [s]80010001200140016001800 T [ K ] Elmer: Temperature profile at x= 5.0mm0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t [s]0.00.20.40.60.81.0 X (cid:1) [ (cid:2) ] Elmer x= 4.5mmSimulation x= 4.5mm0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0t [s]800100012001400160018002000 T [ K ] Elmer: Temperature profile at x= 4.5mm
Figure 5: Experimental measurements for β -phase evolution in a welding process of Ti-6Al-4V by Elmer et al. [34]along with the corresponding simulation results based on the proposed microstructure model and the identified parameterset θ diff ,β = θ ∗ diff ,β .dissolution of the α s -phase in the case of X α s > X eq α can be successfully modeled by a diffusional approach for α s -nucleation by scaling k α s with a factor of f ∗ = 3 . and selecting c ∗ β = 11 . . We will use this approach in thesubsequent numerical demonstrations. To validate the calibrated microstructure model, we will now compare the predicted microstructure evolutions withfurther experimental data. Another common experimental approach for the characterization of microstructural phaseevolutions are the so-called continuous-cooling transformation (CCT) experiments [33, 50]. Here, the microstructuralprobe is again equilibriated at a temperature above T α s , sta = 1273 K , such that X β = 1 . . Afterwards, the probe iscooled down to room temperature T ∞ = 293 . K at different cooling rates ˙ T CCT . Note that the true cooling rate ˙ T ( t ) is time-dependent and ˙ T CCT is only a representative descriptor. Eventually, the evolving microstructure of these coolingprocedures is recorded on the t × T -space. In contrast to TTT-experiments, the dynamics of the cooling process itselfand the thereof resulting microstuctural phase transformations are now the main aspect of these experimental procedures.Hence, the particular temperature profile of the cooling process is now of great importance and has a large impact onthe emerging phases. The true cooling rates ˙ T ( t ) in practical CCT-experiments are changing over time, being highest inthe beginning of the cooling procedure and tending to zero for infinite long times. Following the procedure in [33],the characteristic cooling rate ˙ T CCT is in the following defined as the cooling rate at ◦ C , respectively . K PREPRINT - J
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15, 2021 (cid:16) ˙ T CCT := ˙ T (cid:12)(cid:12) T =1173 . K (cid:17) . A schematic CCT-diagram for Ti-6Al-4V based on characteristic experimental coolingrates taken from [33] as well as the prediction by the proposed microstructure model are shown in Figure 6. Schematic CCT-diagram for X α (Ahmed et al.) -1 t [s]400600800100012001400 T [ K ] T (cid:1) m ,sta = 848KT (cid:1) s ,end = 935KT (cid:1) s ,sta = 1273K . . . . Simulated CCT-diagram for X (cid:0) T CCT = 525 K/sT
CCT = 410 K/sT
CCT = 20 K/sT
CCT = 1.5 K/s
Figure 6: Continuous-cooling transformation (CCT) diagram for X α in Ti-6Al-4V with characteristic cooling rates ˙ T CCT : schematic diagram adapted from [33] (left) and predicted diagram via proposed microstructure model (right).In [33] two distinct CCT-cooling rates ˙ T CCT are reported which divide the evolving microstructure for continuousTi-6Al-4V cooling into three characteristic regimes. For cooling faster than ˙ T CCT = − K/s , only martensitictransformation was observed. For cooling between ˙ T CCT = − K/s and ˙ T CCT = − K/s , coexisting Martensiteand stable α s transformation were found. These coexisting α -morphologies are sometimes also described as massive- α [4]. Eventually, for cooling rates slower than ˙ T CCT = − K/s only stable α s emerges. In the regime of fast coolingrates, the transformation of Martensite starts at temperatures below the Martensite-start-temperature T α m , sta . For slowercooling rates stable α s -nucleation is the dominating effect, which can already be observed at higher temperatures.From Figure 6 it can already be concluded that the characteristic kink in the isoline X α = 0 . at ˙ T CCT = − K/s ,which separates the pure Martensite from the massive- α regime, is predicted very well by the proposed model. A moredetailed discussion and comparison will be presented in the following sections. In the CCT-simulations presented in the subsequent section, we followed the experimental set-up and results presentedin [33]. In their work, the authors realized different cooling rates through convectional air cooling for low and moderatecooling rates and through water quenching for high cooling rates. The material probe in [33] was a Ti-6Al-4V Jominyend quench bar [51] of . mm diameter cooled down by the mentioned media only at the front face side and thermallyisolated at all remaining surfaces. We want to note that analogous CCT-investigations for steel, based on the numericalsimulation of Jominy end quench tests, have been conducted in the past [52], nevertheless using a simpler microstructuremodel. To mimic the temperature evolution of the actual experiments, we approximated the temperature profile by theanalytic solution for a semi-infinite solid body (sib) under surface convection, which is given in [53]: T sib ( x, t ) = ( T ∞ − T ) · (cid:20) erfc (cid:18) x √ α · t (cid:19) − exp (cid:0) g · x + g · α · t (cid:1) · erfc (cid:18) x √ αt + g · √ αt (cid:19)(cid:21) + T (24)Here, erfc denotes the complimentary error function, T ∞ is the temperature of the cooling fluid and T the initialtemperature of the solid according to [33]. The variable x indicates the distance coordinate measured from the cooledfront surface of the solid and t is the time since initiation of the cooling process. The parameter α describes the thermaldiffusivity of the material, chosen according to α = 10 mm s as an averaged value for Ti-6Al-4V based on [54]. Theparameter g = h/k defines the ratio of the convective heat transfer coefficient h on the surface of the solid to itsthermal conductivity k . Since the specific heat transfer coefficient h from the experiment is unknown, we determine theparameter g in the following in an inverse manner to optimally match four exemplary cooling curves provided in [33].We relax the original assumption of g = const. underlying the analytic solution (24) slightly by introducing a quadratictemperature-dependence for g ( T ) . As discussed above, to make the numerically predicted microstructure evolutioncomparable to the experiments, the model requires cooling curves that match the cooling behavior from the experiments PREPRINT - J
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15, 2021 in good approximation. Considering a potentially temperature-dependent parameter g ( T ) allows us to represent theexperimental cooling curves with higher accuracy. Also from a physical point of few it seems reasonable that g ( T ) isnot necessarily constant across the large temperature spans relevant for these cooling/quenching experiments: g ( T ) = s g · (cid:32) a g + b g · T − T ∞ T ∞ + c g · (cid:18) T − T ∞ T ∞ (cid:19) (cid:33) (25)Based on the experimental set-up as described in [33], we can directly set T = 1323 K. Furthermore, as the continuouscooling of the Ti-6Al-4V probes was described to be conducted with air or water [33], we set T = 293 . K, assumingroom temperature for the cooling fluid. The cooling rates in experiments can typically be adjusted by either differentcooling air speeds or switching to water as a cooling medium for the extreme case of quenching. In order to keep thenumber of unknown parameters limited, we only considered air cooling in the numerical realization of the coolingcurves. To vary the cooling rates as required for the CCT-experiments in the next section, we will adjust the coolingrate by means of an additional scaling parameter s g , which can be interpreted as a manipulation of the cooling airflow velocity. For the inverse parameter identification in this section, we fix the scaling parameter to the default value s g = 1 . The remaining parameters θ thermo = [ a g , b g , c g ] T in (25) are specific to the performed experiment and areinversely determined based on four temperature curves (a-, b-, c- and d-curve in Figure 7) experimentally measured atfour different positions x along the bar axis at otherwise identical cooling conditions [33]. We use again the maximum T [ K ] Ahmed experimental a-curveAhmed experimental b-curveAhmed experimental c-curveAhmed experimental d-curveCalibrated semi-inf. model a-curveCalibrated semi-inf. model b-curveCalibrated semi-inf. model c-curveCalibrated semi-inf. model d-curve
Figure 7: Measured continuous cooling curves by Ahmed et al. [33] along with the corresponding model-based curvesresulting from the identified temperature-dependent parameter g ( T ) from Equations (25) and (24). The a-, b-, c- andd-curve were measured at positions x a = 3 . mm , x b = 9 . mm , x c = 12 mm and x d = 15 . mm at otherwiseidentical cooling conditions.likelihood (ML) point estimate θ ∗ thermo as an optimal choice for the identified parameters required in Equations (24) and(25). Under the assumption of Gaussian measurement noise, the ML point estimate is found to be the minimizer of thesquared loss function over all temperature curves: θ ∗ thermo = argmin θ thermo (cid:20) (cid:90) ( T exp.,a ( t ) − T sib ( x = 3 . , t, θ thermo )) dt + (cid:90) ( T exp.,b ( t ) − T sib ( x = 9 . , t, θ thermo )) dt + (cid:90) ( T exp.,c ( t ) − T sib ( x = 12 , t, θ thermo )) dt + (cid:90) ( T exp.,d ( t ) − T sib ( x = 15 . , t, θ thermo )) dt (cid:21) (26)In Equation (26), T exp.,a ( t ) , T exp.,b ( t ) , T exp.,c ( t ) , T exp.,d ( t ) refer to the experimental temperature measurements in [33]and T sib ( x = 3 . , t, θ thermo ) , T sib ( x = 9 . , t, θ thermo ) , T sib ( x = 12 , t, θ thermo ) , T sib ( x = 15 . , t, θ thermo ) are the corre-sponding simulated temperature profiles using Equations (24) and (25). It is emphasized that all four experimentaland corresponding model-based temperature curves (differing only in the measurement/evaluation position x ) areconsidered simultaneously in the inverse analysis. The broader data basis resulting from this combined approach is apre-requisite for a robust inverse identification procedure and an accurate representation of the experimental coolingcurves. We utilize again the Levenberg-Marquardt optimizer [48] to solve for θ ∗ thermo in Equation (26), which resultsin the identified parameter set θ ∗ thermo = [ a g , b g , c g ] T = [73 . , − . , . T m − . Figure 7 shows a good agreement ofthe analytic model with the temperature measurements by Ahmed et al. [33]. Equation (24) based on the identified PREPRINT - J
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15, 2021 parameters can now be used to generate arbitrary cooling curves for the simulation of CCT-experiments by varying thescaling parameter s g in (25), as shown in the following section. For the simulation-based creation of CCT-diagrams, we chose an equidistant range of scaling parameter values s g suchthat 150 cooling curves between ˙ T CCT = − K/s and ˙ T CCT = − K/s were realized. Afterwards, the evolvingmicrostructure was again recorded over the t × T -coordinate space in form of contour-lines for the individual phasefractions. Figure 8 displays the results of the CCT-simulations, along with four characteristic cooling curves analyticallyreproduced by means of (24) as well as the aforementioned experimental a-curve from [33]. Note that in contrast to the -1 t [s]400600800100012001400 T [ K ] T (cid:1) m ,sta = 848 KT (cid:1) s ,end = 935 KT (cid:1) s ,sta = 1273 K . . . . . . CCT-diagram for X (cid:0) s T CCT = 525 K/sT
CCT = 410 K/sT
CCT = 20 K/sT
CCT = 1.5 K/sAhmed: a-curve 10 -1 t [s] T (cid:1) m ,sta = 848 KT (cid:1) s ,end = 935 KT (cid:1) s ,sta = 1273 K . . . . . . CCT-diagram for X (cid:0) m T CCT = 525 K/sT
CCT = 410 K/sT
CCT = 20 K/sT
CCT = 1.5 K/s 10 -1 t [s] T (cid:1) m ,sta = 848 KT (cid:1) s ,end = 935 KT (cid:1) s ,sta = 1273 K . . . . CCT-diagram for X β ˙ T CCT = −
525 K/s ˙ T CCT = −
410 K/s ˙ T CCT = −
20 K/s ˙ T CCT = − Figure 8: CCT-diagram for the α s -, α m - and β -phases resulting from the proposed microstructure model for theidentified parameter set θ ∗ thermo along with four characteristic cooling curves according to (24) and the experimentala-curve from [33]. Left: contour-lines for X α s . Middle: contour-lines for X α m . Right: contour lines for X β .TTT-diagrams in Figure 3, the isolines in Figure 8 show the true/absolute phase fractions such that the phase fractionvalues of all three plots would sum up to one for any given data point. In [33] it was found that cooling rates fasterthan ˙ T CCT = − K/s resulted in a fully martensitic transformation of the microstructure while cooling rates below ˙ T CCT = − K/s led to a fully diffusional α s -formation. According to Figure 8, the results predicted by our modelare in very good agreement with these findings as the contour-line for X α s = 1% coincides almost exactly with thecooling line of ˙ T CCT = − K/s (see dashed orange line in Figure 8, left) meaning that the proposed model predicts afully martensitic transformation for cooling rates faster than ˙ T CCT = − K/s . We want to emphasize that the inverseidentification of the microstructure model parameters was solely based on the previously presented TTT-data and not onthe CCT data. Thus, the accurate prediction of the CCT diagram confirms the validity of our microstructure model.In contrast to existing approaches, the proposed microstructure model does not explicitly enforce this critical coolingrate as transformation criteria, instead a lower formation bound at roughly ˙ T = − K/s emerges naturally form thedynamics of the diffusion equations, which is a very strong argument for the generality and consistency of the proposedmodeling approach. Furthermore, the predicted results show also a fully diffusional α s -formation (i.e. X α m is close tozero) for cooling rates slower than ˙ T CCT = − K/s (see dashed green line in Figure 8), which is also in very goodagreement with the experimental observations by Ahmed et al. [33]. Besides these characteristic quantitative values,also the overall qualitative appearance of the predicted CCT-diagrams matches theoretical presentations in [33, 50] verywell.
In the context of SLM process simulation, macroscale thermo-mechanical models are typically applied to predict thetemperature evolution, residual stresses and dimensional warping [55–70]. In the following, an advanced macroscale PREPRINT - J
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SLM model of this type [59, 61] will be applied to provide the temperature field during SLM as required by the proposedmicrostructure model. We will first introduce the (thermal part of the) SLM model and the numerical set-up, thendemonstrate the microstructure evolution during SLM for selected points of the domain over time and eventually showsnapshots of micro-structural states for cross-sections and different base-plate temperatures.
The microstructure model is implemented in the parallel, implicit finite element code
Diablo [71]. The evolution of themicrostructure is driven by a one way coupling of the emerging thermal field. The latter is the solution of the balance ofthermal energy, and the associated boundary and initial conditions: ρc p ˙ T = − div q + r, in Ω , (27a) T ( x T , t ) = ¯ T , for x T ∈ Γ T (27b) q ( x q , t ) = ¯ q · n , for x q ∈ Γ q (27c) T ( x ,
0) = T , on Ω ∪ ∂ Ω , (27d)where Γ T is the portion of the total boundary ∂ Ω subject to essential boundary conditions, and Γ q is the portion of thetotal boundary subject to natural boundary conditions. The constitutive behavior is characterized by a temperature-dependent Fourier conduction of the form q = − k grad T, (28)where k is a second-order tensor of thermal conductivities, which may also be a function of spatial coordinates. For thecurrent problem, the heat source r in Equation (27a) plays an important role, that being to represent the deposition oflaser energy into a powder. While there exist various models in the literature, the model in the current implementationwas taken from [72] in an effort to provide the most natural description of the physical process.The numerical implementation consists of an iterative nonlinear solver that uses consistent Fréchet derivatives, with thesolution computed on first-order finite elements via an iterative method to solve the linear system of equations. Timeintegration is performed using the generalized trapezoidal rule. The code uses distributed memory parallelism to speedup the solution of the spatial problem. The solution of the thermal field is evaluated at the Gauss points of the finiteelement discretization. For a full description of the model and its implementation, see [59, 61]. While Diablo containsadditional physics, including solid mechanics and mass transfer, a detailed description is omitted at this point and wedirect the interested reader to [71].In the numerical demonstration, we are investigating the selective laser melting process of a one-millimeter-sided cube onto a base-plate subject to different values of constant Dirichlet boundary conditions T bp ∈ { K, K, K, K, K, K } (which are applied solely on the “bottom” of the base-plate, Γ T = min ( z ) ) and the associated effect on the emerging microstructure distribution. The domain is mostlyinsulated, with the exception of the free surface ( Γ q = max ( z ) ), for which there exists a Neumann boundary conditionthat accounts for the energy loss due to both radiation and evaporation. The initial condition is defined as T = 303 K .The processing parameters consist of a laser travelling at mm/s , with a power of W , beam radius of µm ,and track spacing of µm , with eight laser tracks per layer. The scanning direction is in each subsequent layer rotatedby 45 degrees counterclockwise. The powder layer depth is µm , such that the final cube consists of 34 layers. Thephysical time of investigation is t ∈ [0 s, s ] , with the active processing time occurring over the interval [0 s, . s ] .The geometric set-up and one snapshot of the emerging temperature field and phase variable are depicted in Figure 9.These problems were run using 128 cores, and typically took around 15 to 20 hours of wall clock time to complete. Thesignificant resources needed to simulate laser powder bed fusion (LPBF) problems depend on several factors, includingits multiscale nature, which can be relevant with respect to both the spatial and temporal domains. The spatial scalesare resolved via the use of h -refinement, with refinement indicators tailored to the LPBF problem, as can be seen inFigure 9. With respect to the temporal scales, is it noted that the large number of time steps (associated with the activesimulation time and the high speed of the laser) necessitates more sophisticated methodological treatment in order toreduce the wall clock run times to the useful range without significantly degrading the solution accuracy. Both thegeometric and temporal scale issues are currently being addressed by the Diablo development team, as addressed in themanuscripts [73, 74]. As an example, a cube with cm edges, run with fully nonlinear behavior and physically relevant process parameters, requiresaround 500 million time steps. PREPRINT - J
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Figure 9: Representative state of the thermal model. The top frame depicts the mesh, for which refinement is mediatedby distance from the free surface, the bottom left frame depicts the temperature field, and the bottom right framedepicts the internal phase variable (with gray representing the baseplate, which is consolidated material throughoutthe simulation) as well as containing several temperature contours, with blue being T = 505 K , red being the melttemperature, and yellow being the vaporization temperature. We start the part-scale demonstrations by analyzing the microstructure evolution over time at the cube’s center point P = [0 . , . , . T mm . The layer that contains this point is processed at t ≈ . s . Figure 10a to 10f depict therespective temperature profiles along with the temperature rates and the corresponding microstructure evolution for allsix investigated base-plate temperatures. The temperature profile is supported by characteristic temperatures such asthe Martensite-start temperature T α m , sta , the α s -end and start temperatures, T α s , end and T α s , sta as well as the solidustemperature T sol and liquidus temperature T liq .For T bp < K , respectively in Figure 10a to 10c, we can observe the formation of Martensite directly after thefirst melting by the laser in the time period . s < t < s and then again slightly before t = 4 s , followed by asubsequent stabilization of the Martensite phaseThe amount of formed Martensite phase for t > s is dependent on the metastable Martensite equilibrium phasefraction X eq α m in Equation (5) and decreases with rising T bp . As stated in Equation (6), the metastable Martensite phasewill transform to α s for t → ∞ . Nevertheless, the diffusion rate for X α m transformation is at lower temperatures( T < K ) so small (see Figure 4) that Martensite dissolution, respectively the formation of stable α s remainsunnoticeable. PREPRINT - J
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15, 2021 T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10a: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 303 K . At a base-plate temperature of T bp = 900 K (see Figure 10d) no Martensite formation is recorded as the center node’stemperature stays above the Martensite start temperature T α m , sta but below the α -transus end temperature such that T α m , sta < T bp < T α s , end . The latter condition will result in the maximum amount of stable α s -transformation as the α -equilibrium phase fraction X eq α = 0 . at this temperature is still the same as at room temperature. Figure 10d onlyshows the first 13 seconds of the SLM process but the α s -nucleation will continue until X α s = X eq α = 0 . . Other pointswithin the cube might nevertheless experience a martensitic transformation when the steady-state temperatures fallbelow the Martensite start temperature T α m , sta .For a base-plate temperature of T bp = 1100 K (see Figure 10e), which fulfills T α m , sta < T α s , end < T bp < T α s , sta , weobserve a similar characteristic but at slower α s -formation rates and a lower equilibrium phase fraction X eq α < . .Eventually, for a base-plate temperature of T bp = 1300 K (Figure 10f), which is above T α s , sta , the microstructure at P stays in the β -regime for all times (as long as the base-plate temperature is held at T bp = 1300 K ) and no α s - or α m -transformation is observed.We can summarize the following points for the microstructure evolution at the cube’s center node P :• Without preheating of the base-plate, respectively for moderate preheating ( T bp < T α m , sta ), the Martensitephase is dominating the microstructure in the deposited material.• For T bp ≤ K the center point experiences six liquid-solid transitions that are followed by martensitictransformations. For higher base-plate temperatures T bp > T α m , sta the material melts more often but themartensitic transformation is not triggered anymore. The visible temperature peaks in Figures 10a to 10fconsist actually of several very closely packed peaks that are caused by the neighboring laser tracks that heat upthe center node P again. This process happens so fast (the scanning speed in the simulation was mm/s )that these peaks are not distinguishable on the presented time-scale. The 17 visible peaks corresponds to thenumber of layers that are processed above the center node P such that the cube consists in total of 34 layers.• The absence of martensitic transformation for T bp ≥ T α m , sta in combination with slower formation dynamicsof the stable α s -phase is especially interesting from a mechanical perspective. The β -phase is known to besofter than the α s - or α m -phase [75], such that build-up of residual stresses is expected to be smaller if themicrostructure is dominated by the β -phase for longer times.• Theoretically, Martensite will dissolve for t → ∞ but the transformation will only be noticeable for T > K . PREPRINT - J
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15, 2021 T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10b: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 500 K . T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10c: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 700 K . PREPRINT - J
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15, 2021 T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10d: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 900 K . T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10e: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 1100 K . PREPRINT - J
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15, 2021 T [ K ] T liq T sol T α s,sta T α s,end T α m,sta -1 |˙ T |[ K / s ] X [ − ] X α s X α m X β X liq Figure 10f: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its center point P whichis processed at t ≈ . s with base-plate temperature T bp = 1300 K . Figure 10: Simulated microstructure evolutions during the SLM process of a one millimeter-sided cube at its centerpoint P which is processed at t ≈ . s . The sub-figures show the evolving microstructure for different base-platetemperatures T bp ∈ { , , , , , } K . In this section, we continue the investigations of the evolving microstructure in the SLM example of the one millimetersided cube. In contrast to the former local analysis of the center node, we now want to investigate the resultingmicrostucture distributions within a complete cross-section in the vertical x-z-center plane that contains the center node P of the cube, for several selected snap-shots in time. We furthermore consider the exemplary base-plate temperatures T bp = 303 K and T bp = 900 K to compare the room-temperature case with the first preheated case that does not resultin martensitic transformation. Snapshots are taken for the physical times t ∈ [3 , , s and phase fractions X α s , X α m and X β are depicted in Figures 11a to 11d.For a base-plate at room temperature ( T bp = 303 K ) we observe spatially strongly heterogeneous microstructuredistributions during the ongoing SLM process (see Figure 11a and 11b). The Martensite phase in these cases ispropagating upwards (in positive z-direction) starting from the base-plate towards the last processed layer.When looking at the results for the two investigated cases at t = 15 s (see Figure 11c and 11d), we find a ratherhomogeneous microstructure distribution in both cases with only a slight deviation for the case T bp = 900 K . For T bp = 300 K , we observe an almost complete Martensite transformation. On the contrary, for T bp = 900 K noMartensite is formed. Note, that for this case the phase transition from α s to β is still ongoing (until an equilibriumphase fraction of X α s ≈ . is reached) at the depicted snap shot at t = 15 s . Moreover, we notice a slightly higher α s -concentration at the bottom of the cube, i.e. a higher amount of β has already transformed into α s at t = 15 s , whichis caused by the higher base-plate temperature that leads to higher nucleation rates k → α s ( T ) (see Equation (15)).Coming back to the case T bp = 300 K , it is emphasized that for larger geometries we would in general expect a moreheterogeneous microstructure state due to a more heterogeneous temperature distribution with strong spatial gradientsat the surfaces of the geometry. An extended dwelling time at elevated temperatures, due to the increased thermalmass of larger parts in combination with an increasing amount of absorbed laser energy, would result in longer timesat a temperature band ( T α m , sta < T < T α s , sta ) favoring the diffusional dissolution of α m into α s in the core regionof such parts, while the higher cooling rates and lower temperature levels at the free surfaces are expected to fosterremaining Martensite phase fractions. In the next example, a component of larger size will be analyzed to verify theseconsiderations. PREPRINT - J
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15, 2021 z [ mm ] x [ mm ] x [ mm ] x [ mm ] Figure 11a: Simulated microstructure distributions for a SLM process of a one millimeter-sided cube within its vertical x-z-centerplane that contains the center node P . The microstructure state is shown for a base-plate temperature of T bp = 303 K and theprocessing time t = 3 s . The gray area above marks the layers that are not yet processed by the laser at the time t = 3 s . In the rightfigure, which depicts the β -phase fraction X β , the melt-pool is indirectly visible through the decreased β -phase fraction in the rightcorner of the up-most layers due to the laser that has just previously scanned this plane in x-direction from left to right. In a similarfashion, the decreased amount X β in the upper left corner of the right Figure stems from the heat of the laser that is currently meltingthe subsequent track with a now ◦ rotated scanning direction (diagonal scanning), starting in a corner of the cube. z [ mm ] x [ mm ] x [ mm ] x [ mm ] Figure 11b: Simulated microstructure distributions for a SLM process of a one millimeter-sided cube within its vertical x-z-centerplane that contains the center node P . The microstructure state is shown for a base-plate temperature of T bp = 303 K and theprocessing time t = 4 s . Martensitic transformation propagates from the bottom of the cube to its top, which can be seen by thehigher X α m -phase fractions towards the bottom of the cube in the left figure. x [ mm ] x [ mm ] x [ mm ] z [ mm ] Figure 11c: Simulated microstructure distributions for a SLM process of a one millimeter-sided cube within its vertical x-z-centerplane that contains the center node P . The microstructure state is shown for a base-plate temperature of T bp = 303 K and theprocessing time t = 15 s . A homogeneous microstructure state with full martensitic transformation ( X α m = 0 . ) can be observedthroughout the cube. PREPRINT - J
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15, 2021 x [ mm ] x [ mm ] x [ mm ] z [ mm ] Figure 11d: Simulated microstructure distributions for a SLM process of a one millimeter-sided cube within its vertical x-z-centerplane that contains the center node P . The microstructure state is shown for a base-plate temperature of T bp = 900 K and theprocessing time t = 15 s . Due to the preheated base-plate no Martensite is formed. The diffusional phase transition from α s to β isnot finished at the depicted snap shot. Figure 11: Microstructure distributions within the vertical x-z-center plane that contains the center node P of the onemillimeter-sided cube. The sub-figures show the resulting microstructure state of the phase-fractions X α s , X α m and X β at different times of the SLM process and or two different base-plate temperatures T bp = 303 K and T bp = 900 K . As SLM simulations with consistently resolved laser heat source for larger geometries (one centimeter sided cubesand above) are still hampered by the associated computational costs, even when most modern HPC systems areconsidered, we now want to investigate geometry-scaling effects and the effect of the increased thermal mass on themicrostructure through rapid convective cooling of a 100 mm - sided cube. The numerical examples do not intend toquantitatively mimic SLM processes but rather demonstrate qualitative microstructural characteristics that evolve forparts of practical relevant size under rapid cooling or quenching. These kind of heat treatments have broad applicationsin metal processing.We model a preheated cube at T = 1300 K which is in thermo-mechanical contact with a base-plate with a Dirichletboundary condition at the bottom of first T bp = 300 K and then in a second investigation of T bp = 900 K (see Figure12, red line). The cube is subject to a convective boundary condition with heat-transfer coefficient α c = 1000 Wm K on its free surfaces (Figure 12, blue lines). The surrounding atmospheric temperature is set to room temperature at T ∞ = 300 K . The interface between base-plate and the bottom of the cube is modeled via thermo-mechanical contactinteraction employing a numerical formulation recently developed in [76] and choosing a thermal contact resistancethat is equivalent to an effective heat-transfer coefficient of α tc = 5 · Wm K (Figure 12, bold black line). Theremaining free surfaces of the base-plate are assumed to be adiabatic (see Figure 12, green lines). Note that the orderof magnitude of the heat transfer coefficient α c = 1000 Wm K at the free surfaces of the cube was chosen to mimicconvective cooling, e.g., due to forced air flow. As we only want to demonstrate qualitative results here, we omita further detailed description of a specific cooling scenario. These quenching simulations were conducted with ourin-house multi-physics framework BACI [77].As a consequence of the convective cooling and the thermo-mechanical contact, the cube starts cooling down untilthermodynamic equilibrium is reached. We simulated the microstructure evolution for this cooling process andinvestigated the resulting steady-state microstructure state at t = 5000 s . Figure 13 shows the resulting crystallographicdistribution for a base-plate temperature of T bp = 300 K and Figure 14 demonstrates the result for T bp = 900 K .For both investigated cases, we see a several millimeter strong Martensite coating, which is especially pronouncedat the corners of the cube that are characterized by higher temperature rates and lower temperature levels. Thesequalitative findings are in good agreement with experimental results for quenching of Ti-6Al-4V as well as additivelymanufactured parts [4, 78]. In case of T bp = 300 K (Figure 13) we observe an even higher Martensite amount on thebase-plate-cube interface due to the higher heat transfer at the base-plate. This effect is inverted when the base-plate isheated to T bp = 900 K (Figure 14) and stable α s -phase can be found instead of the previous Martensite phase. Similarto the preceding SLM demonstration, a pre-heated base-plate resulted in a significant reduction of metastable Martensitephase for material close to the cube-base-plate interface. The core of the material in both demonstrations (Figure 13 and PREPRINT - J
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15, 2021 (cid:1) c adiabatic thermocontactT bp T (cid:1) =300 K T =1300 K Figure 12: Schematic set-up for cooling simulation of a 100 mm - sided cube on a base-plate. The initial temperature ofthe cube is prescribed to T = 1300 K . The atmospheric temperature is set to T ∞ = 300 K and the free surfaces ofthe cube have a convective boundary condition (blue) with heat transfer coefficient α c = 1000 Wm K . The base-platehas a Dirichlet boundary condition on the bottom (red) of first T bp = 300 K and, in a second simulation run, of T bp = 900 K and is assumed adiabatic on its free surfaces (green). The cube and the base-plate are modelled to bein thermo-mechanical contact (bold black line) using a thermal contact resistance that is equivalent to an effectiveheat-transfer coefficient of α tc = 5 · Wm K . x [ mm ] x [ mm ] z [ mm ] Figure 13: Simulated microstructure distribution of α s - and α m -phases in the vertical center plane of the 10 cm - sidedcube after 5000 s of cooling. The simulation used a thermal heat-transfer coefficient of α c = 1000 Wm K , a base-platetemperature of T bp = 300 K and a thermal contact resistance that is equivalent to an effective heat-transfer coefficientof α tc = 5 · Wm K . PREPRINT - J
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15, 2021 x [ mm ] x [ mm ] z [ mm ] Figure 14: Simulated microstructure distribution of α s - and α m -phases in the vertical center plane of the 10 cm - sidedcube after 5000 s of cooling. The simulation used a thermal heat transfer coefficient of α c = 1000 Wm K , a base-platetemperature of T bp = 900 K and a thermal contact resistance that is equivalent to an effective heat-transfer coefficientof α tc = 5 · Wm K .Figure 14) is composed of stable α s -phase due to the increased thermal mass of the material when compared to ourformer SLM examples with a 1 mm sided cube in Section 4.3.Also for larger SLM-manufactured geometries, Martensite can be expected mostly in proximity to surfaces in form of aMartensite coating, which is in compliance to experimental findings. During the SLM process, temperature rates inregions close to the melt-pool are extremely high such that we would initially expect a purely martensitic transformationas soon as the temperature decreases below the Martensite start-temperature T α m , sta . Nevertheless, the re-heating of thematerial during the processing of subsequent layers and tracks leads to lower cooling rates in subsequent thermal cycleson the other hand, and to an increasing overall temperature niveau within the deposited material. This effect might leadto longer dwelling times with conditions that are suitable for diffusion-based Martensite dissolution into α s -phase suchthat the core of larger SLM processed parts can also be dominated by the stable α s -phase, depending on the specificgeometry and scanning strategy. With the development of more efficient part-scale simulation strategies for selectivelaser melting, we are planning on answering these questions in future investigations. In this work, we proposed a novel physics-based and data-supported phenomenological microstructure model forpart-scale simulation of Ti-6Al-4V selective laser melting (SLM). The model predicts spatially homogenized phasefractions of the most relevant microstructural species, namely the stable β -phase, the stable α s -phase as well as themetastable Martensite α m -phase, in a physically consistent manner, i.e. on the basis of pure energy and mobilitycompetitions among the different phases. The formulation of the underlying evolution equations in rate form allowsto consider general heating/cooling scenarios with temperature or time-dependent diffusion coefficients, arbitrarytemperature profiles, and multiple coexisting phases in a mathematically consistent manner, which is in contrast toexisting approximations via JMAK-type closed-form solutions.Altogether, the model contains six free (physically motivated) parameters determined in a robust inverse identificationprocess on the basis of comprehensive experimental time-temperature transformation (TTT) and transient heating datasets. Subsequently, it has been demonstrated that the identified model predicts common experimental procedures suchas continuous-cooling transformation (CCT) experiments [33] with high accuracy and reflects well-known dynamiccharacteristics such as long term equilibria or critical cooling rates naturally and without the need for heuristictransformation criteria as often applied in existing models.The part-scale simulation of selective laser melting processes with consistently resolved laser heat source is still anopen research questions. In contrast to existing microstructure models that resolve the length scale of individualcrystals/grains, the proposed continuum approach has the potential for such part-scale application scenarios. To thebest of the authors’ knowledge, in the present contribution a homogenized microstructure model of this type has forthe first time been applied to predict CCT- and TTT-diagrams, which are essential means of material/microstructure PREPRINT - J
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15, 2021 characterization, and to predict the microstructure evolution for a realistic SLM application scenario (employing astate-of-the-art macroscale SLM model) and for the cooling/quenching process of a Ti-6Al-4V cube with practicallyrelevant dimensions.In the investigated SLM process, Martensite could be identified as the dominating microstructure species due to theprocess-typical extreme cooling rates and the comparatively small part size considered in the present study. A preheatingof the material (e.g. via a preheated base-plate) resulted in a decreased Martensite formation and higher phase fractionsof the stable α s -phase. In a subsequent simulation, the rapid cooling/quenching process of a larger cube (side length cm ) resting on a cold metal plate and subject to free convection on the remaining surfaces was considered. It wasdemonstrated that the high cooling rates in near-surface domains can lead to a strong martensitic coating of severalmillimeters, a behavior that is well-known from practical quenching experiments. The slower average cooling rates andthe higher thermal mass in this example resulted in a large core domain dominated by the stable α s -phase.In future research work, the proposed microstructure model shall be employed to inform a nonlinear elasto-plasticconstitutive model, thus contributing to the long-term vision of achieving accurate thermo-mechanical simulationsof selective laser melting processes on part-scale. Furthermore, SLM process parameters shall be inversely adjustedto yield specific microstructural distributions and hence desired mechanical properties by deploying novel efficientmulti-fidelity approaches for (inverse) uncertainty propagation [79]. Acknowledgements
This work was partially performed under the auspices of the U.S. Department of Energy by Lawrence LivermoreNational Laboratory under contract DE-AC52-07NA27344. In addition, the authors wish to acknowledge funding ofthis work by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within project 437616465.Especially, we want to acknowledge Robert M. Ferencz for his assistance and fruitful discussions. Finally, we want tothank Sebastian Pröll, Abhiroop Satheesh and Nils Much for their support in model verification.
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