Microstructural Modeling of the α + β Phase in Ti-6Al-4V: A Diffusion-Based Approach
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Complex heat treatment operations and advanced manufacturing processes such as laser or electron-beam welding will see the metallic workpiece experience a considerable range of temperatures and heating/cooling rates. These intrinsic conditions will have a significant bearing upon the microstructure of the material, and in turn upon the thermo-mechanical properties. In this work, a diffusion-based approach to model the growth and shrinkage of precipitates in the alpha + beta field of the common titanium alloy Ti-6Al-4V is established. Further, the numerical model is extended using a JMA-type approach to explore the dependency of the beta-transus temperature on extremely high heating rates, whereby dissolution alone is too slow to accurately describe the alpha to beta-phase transformation. Experimental heat treatments at rates of 5, 50, and 500 °C/s were performed, and metallographic analysis of the samples was used to validate the numerical modeling framework predictions for lamellar shrinkage, while data from the literature has been used to evaluate the numerical modeling framework for the growth of equiaxed microstructures. The agreement between measurements and numerical predictions was found to be good.
Titanium alloys are excellent material candidates for high-performance, extreme service-condition applications. Many aerospace structures including airframes, skin, and engine components have benefited from the use of titanium alloys, where the reduced weight and increased performance permit lower fuel consumption and emissions, factors which are becoming increasingly important. An important factor within any component manufacturing process is the joining method to be used for the alloy of interest. Although there are numerous different joining methods of potential use, welding can produce components that are lighter, have greater structural integrity, and are cheaper than other methods can achieve (e.g., riveting). However, in order to produce a welded joint with the required structural integrity and material properties, one must consider the metallurgical, thermal, and mechanical fields to optimize the outputted joint.
Numerical methods have been developed to describe the thermal, metallurgical, and mechanical phenomena associated with welding; however, the behavior of many materials are so complex that simple analytical solutions are typically insufficient to predict material behavior during a welding process with sufficient accuracy. In particular, mechanical properties of materials are strongly dependent upon the microstructure present, and the rate at which microstructure changes in response to local thermal fields are often complex. A requirement for greater accuracy is being met by numerical methods where the governing thermal, mechanical, and metallurgical equations can be solved over a series of fine volumetric regions, which has led to the potential for location-specific behavioral solutions. The traditional finite element numerical methods are an example of this, and their increasing use gives researchers the chance to further their understanding of these complex interactions during welding and other processes.
Sysweld (ESI Group) is a specialist welding FE code, which contains a microstructural modeling framework based on sub-routines originally developed for steels,[3, 4, 5] where the temporally evolving phase proportion is described by phenomenological equations. However, to accurately predict the mechanical behavior for a titanium alloy such as Ti-6Al-4V, as well as the distribution of phases it is also necessary to predict the morphology of the grains developed during the process, and in particular the dimensions of spherical, lamellar, and acicular grains,[6, 7, 8] to allow for accurate mechanical property predictions.
An improved FE method for predicting metallurgical and mechanical behavior of Ti-6Al-4V when subjected to a high energy density welding process is therefore of considerable interest to the modeling community. A diffusion-based approach is presented to describe alpha-phase evolution in the α + β field, for the equiaxed and lamellar morphologies. This model is integrated with a JMA equation to describe the change of crystallographic structure for conditions close to the beta-transus temperature and with fast heating rates. Both experimental and literature data are used to validate the numerical predictions. An overarching requirement of the metallurgical modeling framework is to generate models capable of being implemented in commercial simulation software and being calculated on a workstation computer without significantly increasing the computational time.
Material and Methods
Measured Composition of the Ti-6Al-4V Nominal Plate
To study the dissolution of the microstructure during heat treatments, the 6-mm-diameter samples were subjected to different isochronal heat ramps using the digital closed loop control thermal and mechanical testing system Gleeble 3500, (Dynamic Systems Inc). To allow the Gleeble machine to control the temperature at which the samples were subjected during the tests, three type-B thermocouples were spot welded on to the specimens, at the mid-length location and at 4 and 8 mm from the mid-length, respectively. The Gleeble system produces a parabolic temperature distribution across the sample, so three different heat treatment results could be obtained from each specimen.
In the literature, a common procedure adopted for lamellae thickness measurement consists of collecting a series of intercept readings by rotating a grid of parallel lines at many angular steps, till a complete rotation of 360 deg has occurred. The mean length of all intersections of the regular grid, for each lamellae, multiplied by a factor 1.5 and inverted, would return the hypothetical true mean value of the lamellae measured. However, this approach would have only allowed for lamellae thickness measurements.
Thus, to estimate the evolution of the lamellae in two-dimensions, their width and length in the as-received microstructure have been characterized. Ellipses were manually inscribed with the major axis coincident with the length of the lamellae, and the minor axis coincident with the thickest section of the lamellae considered (Figure 1). The mean lamellae thickness was calculated, and by the ratio of the mean length and mean thickness, the mean aspect ratio was estimated. In this work, only lamellae colonies appearing with sharper boundaries were considered. This reduced the effect of the random inclination of the sectioning plane used. Colonies not sectioned perpendicularly to their longitudinal direction would appear with darker areas close to their boundaries (shadow effects) and, usually, it could be noticed that these lamellae had overall larger interior and boundary thicknesses. The volume fractions of alpha and beta phase were estimated by thresholding image analysis. Detailed image analysis was performed to further understand the microstructural evolution present within samples.
The parent alpha lamellar microstructure during heating dissolves into the beta phase. As the material is subsequently cooled by water quenching (with rates in excess of 300 °C/s), a martensitic microstructure develops in place of the beta phase, if peak temperature and cooling rate are both sufficient. The as-received material has a mean lamellae thickness of approximatively 3.0 μm with a standard deviation of 0.5 μm. The samples were tested at heating rates of 5, 50, and 500 °C/s.
The kinetics of the diffusion process require a dwell time to allow the metallurgical transformation to occur. In a sufficiently fast heating process, there is insufficient time above the beta-transus for complete dissolution of the lamellae before quenching: their thickness at temperatures close to the beta-transus can remain quite large and the solid-state crystallographic change from HCP to BCC occurs before their complete dissolution. Additionally, concentrations of the constituent elements within different phases do not have time to change to the equilibrium concentration. This is particularly visible when the peak temperature only just exceeded the beta-transus temperature, and the brighter (beta) and darker (alpha) fields associated with the as-received microstructure in the lamellar morphology can be still observed.
Mathematical Modelling Framework
Field and Flux Balance
Where possible, the exact solutions of the field equation (Eq. ) and flux balance (Eq. ) have been used; otherwise, simplified conditions have been applied. The concentration field in the matrix is assumed, during growth, to deplete the matrix immediately ahead of its interface[16,17] since a solute is needed to increase the particle dimensions. Conversely, during shrinkage, the precipitate rejects solute that moves toward the matrix, resulting in a gradual decrease in the solute distribution from the particle to the matrix.
The assumptions involved for the calculation of CM in Eq.  neglect the precise solute concentration in the region adjacent to the precipitate. This simplification introduces a small 0.5 to 1.5 pct under-prediction of the volume fraction of primary alpha transformed. Phase field calculations support this hypothesis, in particular for long annealing times.
All the terms have been already explained previously. The complementary error function (erfc) has the form as given in Eq. .
Lamellar α-Phase Particles
The lamellar particle growth has been modeled using the solution proposed in References 13, 22, and 23 for an ellipsoidal shape, whose aspect ratio between minor and major axis remains constant. In Reference 9, it has been stated that this solution gives better results than a semi-infinite plate solution, where the growth of the particle seems to be underestimated. However, this approach does not consider the initial stage of the growth of lamellar particles, where after nucleation a lengthening phase follows until lamellae mechanically impinge on each other. Since nucleation is out of scope for this model, the assumption of a constant aspect ratio is assumed to be reasonable for the description of growth and shrinkage in the α + β field. Nucleation and lengthening combined could be considered using a further analytical model.
All the considerations for the calculation of the inter-diffusion coefficient D were considered valid for all the models presented in this work. The intrinsic diffusion coefficients reported in Eqs.  through  were corrected by the thermodynamic factor (Eq. ). The resulting inter-diffusion coefficient is dependent on the temperature and concentration of the beta matrix at the precipitate/matrix interface.
Spherical Precipitate Growth
To validate the model, data in the literature[9,11, 12, 13, 14] on the study of the spherical particle growth have been used. Thus, the microstructural measurements prior to heat treatment from the relevant literature sources were used as the initial conditions for the numerical model: initial average radius of the spherical particles 4.5 μm, initial phase fraction of spherical particles 0.27, material composition (in weight percent) of 6.4 aluminum, 4.2 vanadium, 0.14 iron, 0.19 oxygen, 0.016 carbon, 0.004 hydrogen, 0.005 nitrogen, and the balance titanium. The alpha-phase fraction of 0.27 was obtained by soaking the material at 955 °C for 1 hour and then water quenching it. The heat treatments tested consisted of cooling at 3 different cooling rates (11, 42, and 194 °C/min) starting from a soaking temperature of 955 °C after a 20-minute hold.
The CP value in Eq.  has been set as the equilibrium concentration of the diffusing element in the alpha phase, at the starting temperature of the heat treatment, and it was kept constant during the heat treatment. This assumption gives negative values of the supersaturation for the first stage of the heat treatments because the initial alpha-phase fraction (0.27) is higher than the equilibrium value for 955 °C and, consequently, the particle tends initially to reduce its dimension.
The initial soaking prior to heat treatment is not enough to obtain the equilibrium microstructure at 955 °C. A composition of the particle (CP) corresponding to the equilibrium value at about 900 °C would not give the initial shrinkage but, since Reference 14 does not report the actual initial value of CP, the original solution presented has been preferred. A comparison with a hypothetical model where negative supersaturation values give no shrinkage has been made. In this manner, the impact of the different particle dimensions and relative solute concentrations has been evaluated on the final phase fraction predicted.
Lamellar Precipitate Dissolution
The experimental data taken from the initial lamellar structures present within the microstructure are as follows: (i) aspect ratio: 7.0, (ii) lamellae thickness: 3.0 μm, (iii) alpha lamellar phase fraction: 0.95, (iv) mean vanadium composition in the lamellar microstructure: 2.47 pct, (v) mean aluminum composition in the lamellar microstructure: 5.495 pct, and (vi) overall vanadium concentration in the material: 4.0479 pct. Since the model to describe the lamellar microstructure evolution showed very low sensitivity to small variations of the lamellae aspect ratio, a value of 7 was assumed to be constant for all the thermal treatments tested.
Thus, a complete mathematical framework has been presented for the prediction of both spherical and lamellar particle growth and shrinkage, suitable for detailed microstructural modeling of titanium during a range of heating rate processes. These models could potentially be embedded as sub-routines or sub-models within thermal-mechanical process modeling codes to allow for detailed titanium alloy microstructure predictions.
Predictions for the β-Transus Temperature
Parameters Used for Eq. 
For equiaxed grains, the model predicts the phase fraction evolution during cooling reasonably when considering vanadium as the single diffusing element, and better than with aluminum as the diffusing element. For lower heating rates, this difference is exacerbated, whereas at higher cooling rates both vanadium and aluminum diffusion results converge. The predictions are relatively insensitive to small measurement errors of initial particle size.
For lamellar structures, the model predicts the correct trends for lamellae thickness and volume fraction values, with some error in the thickness calculations being more noticeable for the fastest heating rate. The latter could be due to the high internal energy of the material, allowing a switch to the more stable BCC crystallographic structure even though the dissolution of the alpha phase is incomplete.
In order to determine the beta-transus as a function of the heating rate, a JMA approach was adopted and applied to the diffusion-based model to establish when a complete transition from alpha to beta phase was obtained; this was independent of the state of the alpha-phase dissolution.
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