Introduction

The integration of models across several engineering disciplines, such as materials, manufacturing, quality, design, structural analysis and lifing, can yield significant benefits in the quest for optimal component and system designs. Such an integrated computational framework has been demonstrated under the United States Air Force (USAF)-sponsored foundational engineering problem (FEP) program and other programs to optimize aerospace components for location specific properties that meet design intent [1, 2]. In addition, digital thread efforts, supported by the US Department of Defense and focusing on integrating and analyzing digital representations of data to enhance acquisition and sustainment practices, are rapidly progressing [3]. Several government-funded and industry-led programs are building modeling tools and methods that can be transitioned efficiently into engineering practice. Over the last two decades, programs aimed at developing and demonstrating titanium microstructure and mechanical property models for Ti-6-4 were initiated [4,5,6]. Models were developed to predict the crystallographic texture as a function of location in components using a post-processor-type linkage between the finite element analysis (FEA) package DEFORM and the Los Alamos Polycrystalline Plasticity (LApp) code. These projects were also successful in developing tools for aircraft engine manufacturers and their suppliers to predict key microstructural feature evolution, e.g., chemistry, primary α size and volume fraction, β grain size, grain boundary α thickness, and lamellar α thickness, that could be used as inputs into empirical models that predict tensile and fatigue properties.

Performance of metallic structural components is a strong function of its composition and manufacturing process paths that include several thermomechanical parameters such as temperature holds, heating and cooling rates, strain rates, strain magnitudes. The resulting structure (including texture) distribution in a component controls final location specific properties and performance of the part during service. Figure 1 illustrates this point using an example of a generic forged pancake simulation that shows radial and axial process parameter gradients that could potentially result in significantly different mechanical properties, by location. Test samples extracted from locations shown in Fig. 1 could define a range of material pedigrees, resulting from variability in processing path parameters, in the form of multiple material property definitions based on spatial distributions by specific component locations or volumes.

Fig. 1
figure 1

Generic forging simulation illustrating example thermal and property gradients in the axial and radial directions. Test sample locations are depicted by rectangles and circles

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However, current material specification requirements for parts are traditionally based on a single property minimum determined from test samples extracted at various locations [7,8,9,10]. This approach of determining statistically based minimum material property definitions does not exploit the full potential capability of material properties or support critical design review and assessment of locations of highest risk. In contrast, a model-based material definitions approach uses analytical relationships that link composition, manufacturing processes, structure, and properties to describe property distributions and gradients throughout the volume of a specific part.

A formalized approach linking process–structure–property–performance (PSPP) parameters for Ti-6-4 in the α + β processed condition is shown in Fig. 2. This PSPP formalization helps to illustrate the critical relationships between process steps, structural features, mechanical properties, component analysis, and the required set of submodels for use in an MBMD workflow. This paper describes the industrial application of linked PSPP models that have enabled optimization of processes and location-based properties for enhanced titanium component capabilities.

Fig. 2
figure 2

Process–Structure–Property–Performance (PSPP) diagram for Ti-6-4 demonstrating specific relationships. Red lines indicate key processes that influence microstructure and subsequently properties that have been predicted in this paper

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Overview of Process–Structure–Property Models

The following sections provide a brief overview of the various stages of manufacturing Titanium Ti-6-4 alloy from ingot melting to final forging, along with models that have been developed to predict the evolution of key microstructure features. For detailed information on these models the reader is directed to relevant references provided in this article.

Select Titanium Ingot to Billet Conversion Models

Significant efforts to optimize ingot melting practice to maximize productivity, while maintaining acceptable solidification conditions to ensure metallurgical quality and homogeneity have been made over the last two decades [11,12,13,14]. Utilization of vacuum arc remelt (VAR) ingot melt models to assess and quantify the impact of melt profile changes (e.g., arc current, arc voltage, melt rate, magnetic stirring volume) on pool depth, solidification conditions, and overall chemistry distribution (i.e., macro and micro segregation) is prevalent in industry today. Computational models have also been developed for the cold hearth melting (CHM) process used for titanium alloys like Ti-6Al-4 V [14]. These models are based on governing equations for mass, momentum, and energy conservation, accounting for phenomena like thermal and compositional buoyancy, aluminum evaporation, electron beam power distribution, and resistance to flow in the mushy zone. Such models help optimize melt process parameters and improve overall ingot quality.

Similarly, the use of the commercially available FEA software packages like DEFORM, together with key microstructure and damage models allow for the design and optimization of thermomechanical process stages of titanium products to obtain target material structure requirements. Primary Titanium alloy processing from ingot to intermediate products like billet, bar, plate and sheet includes multiple thermomechanical stages that breakdown coarse β grain structures to equiaxed alpha grains in a transformed beta matrix. Most of the microstructure model development has been concentrated in the β phase recrystallization (RX) and the final two phase (α + β) working stages because of the critical impact to final product structure, texture and properties [15,16,17]. Control of β grain size during the recrystallization stage of processing is dependent on prior two phase (α + β) work (prestrain), heating rates, soak temperature and time at temperature.

β grain size control is very important for formation of α colonies and microtextured regions that impact fatigue properties. However, obtaining a uniform β grain size across the billet cross section is extremely challenging due to the spatial non-uniformity of strain arising from the non-isothermal cogging operation typically used by mill product converters. In addition, the large billet cross sectional size results in significant thermal gradients during heating and subsequent quenching operations, further contributing to challenges associated with obtaining uniform structures. Models for static recrystallization of Ti-6-4 using the Avrami formulation have been reported and used with mixed success [18].

Furthermore, static grain growth can follow recrystallization during the β annealing stage, contributing non-uniform β grain sizes. Most studies on static β grain growth have observed both normal and discontinuous grain growth [19,20,21,22,23]. Discontinuous grain growth behavior, associated with slower rates of grain growth, has been attributed to the effect of texture on grain boundary energy and mobility. Additional work, conducted as part of Materials Affordability Initiative (MAI) programs developed a phase field model to predict texture-controlled β grain growth incorporating the local grain boundary energy density to account for the evolution of texture [24].

The effect of cooling rate from the β working stage and the subsequent subtransus reheating temperature have a significant impact on the size and morphology of α laths, and the thickness of grain boundary α. Both grain boundary α and lamellar α lath width have been shown to be inversely related to cooling rate and empirical relationships have been developed by Brooks, Neal and Fox [25, 26]. During the subsequent heating for subtransus working of the billet, the α lath width coarsening has been modeled based on behavior observed by Lifshitz for solute rich particles in a lean matrix [27, 28].

The final processing stages start with an α lath colony microstructure that is quite stable and cannot be modified to a more equiaxed morphology exclusively through heat treatment alone. Hence, subsequent billet hot working and annealing in the α + β phase field is required to produce spheroidization of the lath structure (via both static and dynamic mechanisms). Several researchers have examined microstructure evolution behavior that is pertinent to this processing stage. For example, Li and Semiatin have reported details on the increase in α lath width with increasing annealing time, which affects the final α spheroidized size [29, 30]. Models for the static spheroidization process were based on work by Stefansson on small-scale compression tests and static heat treatments which produced a linear relationship between the volume fractions spheroidized and log time, where both the gradient and the intercept are functions of strain [31]. Margolin and Semiatin observed high angle α boundaries forming within laths during hot working and postulated that these boundaries lead to segmentation of the α laths (by β phase penetration) with increasing strain [28, 32]. Subsequently, the complex relationship between temperature, strain and strain rate on the kinetics of α spheroidization was further developed and quantified by Semiatin et. al [30, 33]. While, these models provide an average level of spheroidization (local texture variations between colony α laths result in different levels of dislocation accumulation and resulting spheroidization) for relative comparisons, they can provide excellent guidance toward process parameter optimization [15,16,17].

Examples illustrating the use of microstructure models to optimize the cooling rates, billet size and shape, cogging forging bites and drafts to achieve a high-volume fraction of spheroidized microstructure from the critical β recrystallization stage have been reported previously [11,12,13]. These linked models are being used by primary product suppliers to optimize process design sequences to achieve improved microstructure control, and product yield that meets customers’ requirements [13, 25]. In addition, these billet conversion Process–Structure models provide a critical linkage to the next stage of component manufacturing process and component qualification and certification. This could be accomplished by predicting billet microstructure distributions for specific zones located at the center, mid-radius and surface locations and then linking this to final component forging process models that predict location specific component properties. Alternatively, these billet zones could also be measured using suitable characterization methods that provide the resolution required to develop statistically equivalent representative volume elements (SERVEs), that could subsequently be used in the final forging process models for location specific component property predictions. This alternate method is being used for the prediction of critical microstructure features like primary α size and volume fraction through the component processing stages that will be described in the next section.

Final Component Process–Microstructure–Property Models

Following billet processing, numerous critical titanium aerospace products are produced via closed die forging processes typically performed between 50 and 100 °F below the β phase transformation (Tβ) temperature for the specific applied alloy heat chemistry. After these forging steps the part goes through a solution and anneal heat treatment (STA) typically conducted between 40 and 70 °F below the Tβ. Microstructures attained after this process are characterized by equiaxed primary α in a transformed β matrix, that results in a good balance of strength, creep, fatigue and crack growth properties. Mechanical properties of α + β titanium alloys depend on the characteristics of critical microstructure features that span a wide range of length scales and are also interdependent on other structural characteristics. The complexity of microstructural features has made it challenging to develop accurate models for mechanical properties [34, 35].

Over the last two decades OEM and U.S. Air Force Research Laboratory (AFRL) sponsored programs developed and validated titanium process, structure and property models for Ti-6-4 [4,5,6, 34,35,36,37]. A package of tools was developed and linked in DEFORM (a commercial FEA software code developed by SFTC, Columbus, Ohio, USA) to predict location specific texture and microstructure characteristics, in forged and heat treated Ti64 components. DEFORM was enhanced to accept descriptions of α phase fractions as a function of temperature and chemistry (i.e., β phase approach curve) obtained via measurements or the use of commercially available thermodynamic software packages [5, 36, 37]. Heat treatment temperatures, local cooling rates experienced at a given location in the forging along with the starting primary α size and volume fraction are used to predict the final size and volume fraction of primary and secondary α present at that location. These projects successfully validated tools that predict key microstructural features, i.e., primary α size and volume fraction, β grain size, grain boundary α thickness, and lamellar α thickness, that could then be used as inputs into tensile and fatigue property models.

To build a composition and microstructure based tensile property model for α + β titanium alloys, an artificial neural network (ANN) model was developed and is schematically shown in Fig. 3a [4,5,6, 35]. This model was trained across a range of measured compositions and microstructures that are typically encountered in Ti-6-4 components. Following the completion of this work researchers extended these efforts and developed a mechanism based phenomenological model that captures the influence of solid solution strengthening, Hall–Petch boundary strengthening and inherent strengthening effects in an analytical formulation for Ti-6-4 tensile properties [38]. Both property modeling approaches identified solid solution strengthening of the α phase (Al, O) along with the volume fraction and size of primary α as being the significant contributors to the strength of α + β processed Ti-6-4. Microstructure and tensile property model validation completed by P&W, using a combination of measured element compositions and predicted microstructures (shown in Fig. 3a) on various forged components and locations showed excellent correlation with experimental results, Fig. 3b.

Fig. 3
figure 3

Ti-6-4 Tensile model a workflow and b tensile strength (yield strength shown as gray dots and UTS shown as blue dots) validation results bounded by a dashed 5% tolerance band. The black solid line represents a perfect prediction

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For over a century, engineers have been concerned with the safe design of complex structures that are loaded under cyclic stress conditions. Much of the early studies on high cycle fatigue (HCF) were conducted on structural steels undergoing completely reversed loading conditions, where the mean stress, \(\sigma_{{\text{m}}} = 0\). However, subsequent work identified the importance of accounting for mean stress effects on fatigue behavior, which was first represented by Haigh in constant life diagrams [39]. Following which, several design models were developed relating mean stress and strength effects to stress amplitude including the Goodman model that conservatively predicted HCF behavior of over 90% of metals [39, 40]. This relationship can be expressed as,

$$ \sigma_{{\text{a}}} = \sigma_{ - 1} \left( {1 - \frac{{\sigma_{{\text{m}}} }}{{\sigma_{{{\text{ult}}}} }}} \right) $$
(1)

, where \(\sigma_{{\text{m}}}\) is the mean stress, \(\sigma_{{{\text{ult}}}}\) is the ultimate tensile strength, \(\sigma_{ - 1}\) is the alternating stress amplitude at a stress ratio \(R = \sigma_{\min } /\sigma_{\max } = - 1\) and \(\sigma_{{\text{a}}}\) is the stress amplitude. A very large percent of ductile materials data lie above the Goodman model threshold, represented by a straight line between fully reversed (\(R = - 1\)) fatigue data and ultimate strength, is deemed sufficiently conservative for most engineering design purposes.

However, fatigue performance of Ti Alloys depends on several factors such as alloy type (\(\alpha + \beta\) vs. near α Ti Alloys), microstructure, crystallographic texture, mean stress, notch sensitivity, and surface conditions. α + β titanium alloys like Ti-6-4 in the duplex microstructural condition exhibit an anomalous mean stress sensitivity characterized by significantly lower fatigue strength that lie below the Goodman line, as shown in Fig. 4 [40,41,42,43,44]. Ivanova and Cohen investigated the influence of microstructure on mean stress dependence on fatigue behavior of Ti-6-4 in multiple microstructural conditions and observed increasing mean stress dependence with increasing primary α size (> 5 µm) and increasing volume fraction α (> 25%). The anisotropic, HCP α phase has been observed to be the weak link in the structure. Fractography of the failed HCF samples revealed faceted initiations associated with large α grain particles that were typically at the tails of the grain size distribution for the material pedigree [41, 42]. Morphology and orientation of the α phase (i.e., microtextured regions) will determine local strain heterogeneity leading to fatigue initiation and short fatigue crack propagation behavior. It is postulated that the crack nucleation and early crack propagation occurs on basal planes in α grains oriented for easy slip activation [45].

Fig. 4
figure 4

Schematic representation of a constant mean fatigue life diagram showing the linear Goodman model (solid black line) and a typical nonlinear “anomalous” behavior for bimodal Ti-6-4 microstructures (blue circles and dashed curve) at 107 cycles. The effect of stress ratios between R = − 1 and R = 1 on fatigue life is shown by gray reference lines. Data has previously been published [34, 35]

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A statistical microstructure-based model was developed at P&W for fatigue strength of \(\alpha + \beta\) titanium alloys, e.g., Ti64, Ti6246, SP700, for equiaxed and bimodal microstructures with primary α particle size ranging between 3 and 16 µm and volume fraction between 0.15 and 0.80 as a function of life for stress ratios between − 1 < R < 0.8. This fatigue strength model was developed to describe behavior across two life regimes LCF (N ≤ 105), HCF (105 < N ≤ 107) and can be represented as a function of

$$ \begin{aligned} \sigma_{{\text{a}}} = & f({\text{Primary}}\;\;\alpha \;\;{\text{Grain}}\;\;{\text{Size}},\;\;{\text{Primary}}\;\;\alpha \;\;{\text{Volume}}\;\;{\text{Fraction}},\;\;{\text{Ultimate}}\;\;{\text{Tensile}}\;\;{\text{Strength}}, \\ & {\text{Reduction}}\;\;{\text{in}}\;\;{\text{Area}},\;\;{\text{Cycles}}\;\;{\text{to}}\;\;{\text{failure}},\;\;{\text{Stress}}\;\;{\text{Ratio}}) \\ \end{aligned} $$

where \(\sigma_{{\text{a}}} = {\text{Fatigue}}\;\;{\text{strength}},\;\;{\text{and}}\;\;{\text{stress}}\;\;{\text{ratio }} = R = \left( {\sigma_{\min } /\sigma_{\max } } \right).\) This HCF model captures the mean stress sensitivity and the “anomalous” behavior exhibited by Ti-64 very well, as shown in Fig. 5.

Fig. 5
figure 5

Cycles to failure predictions (dotted lines) compared to measurements (individual data points) for Ti-6-4 showing mean stress sensitivity effect with increasing R ratio. Data has previously been published [35, 36]

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The next section will describe the use of these PSPP relationships in MBMD workflows for enhancing component designs, properties, and performance.

MBMD Workflow Paradigm for Enhanced Component Strength and High Cycle Fatigue Performance

Traditional engineering workflows assume a single set of minimum part properties defined over a range of measured properties across a component that result from process path differences. This legacy approach does not utilize full location-based material capability, but instead uses the lowest of minimum measured properties. Property variations across a component are assumed to be resulting from uncertainty in material composition and process parameters, which are not accounted for as specific process path related differences. In contrast, model-based materials definition frameworks have the potential to optimize manufacturing part and process designs to enhance legacy material component properties. Furthermore, the predictive capability of an integrated framework can enable overall component performance to be evaluated rapidly over the application boundary condition space [46].

An MBMD framework for Ti-64 rotor component performance enhancement was developed by employing linked PSPP submodels that were discussed earlier and are shown in Fig. 6. These submodels included thermodynamic calculations of volume fractions of the α phase at heat treat temperatures, and a process model of the component heat treatment process which predicts process path based microstructural features (i.e., size and fractions). These microstructure predictions are then in turn used to predict tensile strength and high cycle fatigue properties at component locations of interest. Uncertainty in key model inputs was incorporated and propagated to final mechanical properties. This workflow enabled the assessment of potentially improved property capabilities as a function of microstructure and process variability, leading to separately zoned material pedigree definitions, for the rotor.

Fig. 6
figure 6

Linked PSPP model workflow with UQ for generating MBMDs

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An important part of the described MBMD workflow is the ability to zone a large component into sub-volumes that describe specific design requirements, which in turn require controlled process paths to produce appropriate microstructures. Statistically equivalent representative volume elements (SERVEs) provide a means to statistically describe key microstructure and microtexture features that could be used to predict mechanical properties such as static and dynamic strength [47, 48]. In this effort, samples were extracted from zones of interest in the component rotor and 2-D scans using scanning electron microscopy (SEM) based backscatter electron (BSE) and electron backscatter diffraction (EBSD) methods were utilized to quantify key microstructural features like primary α gain size. In order to generate microstructure-based SERVEs (M-SERVEs) multiple scan volumes were completed until the convergence of the distributions were observed after scanning about 1000 grains, as shown in Fig. 7. SERVEs were developed for all key microstructure inputs into the property models, e.g., primary and secondary α sizes.

Fig. 7
figure 7

Microstructure variation for primary α grain size as a function of increasing sample size leading to convergence. Colors represent standard deviation between 1 and 3. The dashed vertical line identifies a minimum scan area for converged microstructure features

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A Monte-Carlo probabilistic methodology within crystal ball software was used to account for uncertainty in the tensile and HCF models, automatically calculating results of thousands of scenarios, and providing insight into the critical factors. Statistical distributions (M-SERVEs for microstructure elements such as α grain size) for inputs into the tensile and HCF models are used to perform numerous iterations, each time using a different set of random samples from the input distributions. Results from this analysis of tensile and HCF property models are shown in Figs. 8 and 9. Uncertainty based parameter sensitivity analysis suggests compositional levels of Al and O along with the size of primary and secondary α significantly influence the strength and HCF properties.

Fig. 8
figure 8

Strength model distributions and sensitivities

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Fig. 9
figure 9

HCF model distributions and sensitivity analysis for specific stress ratios, a R = 0.4 and b R = − 1

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Probabilistic tensile and HCF modeling results were used to design a manufacturing process to achieve optimal microstructures comprising 15–50% by volume of primary alpha grains with a grain size less than 10 µm and Widmanstatten secondary α laths with less than 1.0 µm in width in locations near the outer rim surface of the rotor [46]. This was specifically accomplished by manufacturing protrusions in precise locations of the rotor that underwent controlled cooling rates. This MBMD workflow has led to new local material pedigree definitions that are tightly controlled by optimizing the heat treatment geometry and process parameters during rotor manufacturing, Fig. 10 [8]. These local pedigrees are characterized by unique structures and properties that would not have been able to be defined by traditional empirically derived methods.

Fig. 10
figure 10

Schematic of (a) turbine engine rotor illustrating the application of model-based definitions across the (b) cross section with specific structures and processing to deliver local pedigree properties for enhanced capabilities

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MBMD Workflow Paradigm for Manufacturability in Design

During the component design phase, it can be beneficial to assume an MBMD framework by extending the workflow discussed above, to evaluate manufacturability constraints against materials performance requirements as part of the initial hardware design and capability assessment. This approach inverts the previous example of optimizing material performance for a design, and instead defines the objective as evaluating the suitability of a design given known requirements for material performance and the distributions of material structure resulting from available manufacturing process paths. This type of automatable co-design paradigm ensures that manufacturability and its interconnected relationships with material performance are integral to the development of new hardware, mitigating potential future challenges to part quality and process yield.

One such MBMD framework for Ti-6-4 rotor manufacturability and design evaluation was developed and utilized through similar application of previously described PSPP submodels. The primary difference for this use case, however, is that necessary manufacturing methods are determined for producing the part envelope around the desired final geometry. Once this is known, Bayesian probabilistic models of material microstructure for the applicable processing paths can be defined, calibrated to measurement, Monte-Carlo sampled, and then uncertainty propagated to each of the linked PSPP submodels. This process is shown visually in Fig. 11. Note, the key difference with Fig. 6 is a parametric part design, which is used in a co-design approach to determine suitability of the design and its requirements to the material and feasible processing paths, rather than the reverse. In addition, when operating in a Bayesian framework the probability of achieving the desired part performance can be evaluated directly. Input distributions of multiple manufacturing processes are updated and evaluated against the probability of success to meet the design requirements, as shown in Fig. 11 This allows for highly actionable insight into the suitability of a design to the material and processing path, enabling quantitative decision making in the context of estimated risk to meeting design requirements. In this case it was possible to determine with high certainty the compatibility of the intended design with available manufacturing paths for the material, enabling rapid feedback to part designers to evaluate options, as needed.

Fig. 11
figure 11

Linked PSPP workflow of MBMD enabled approach with UQ to determine suitability of design requirements to process options

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This subtle difference in approach requires early consideration of the interconnected nature of hardware design, manufacturing, and materials performance to ultimately provide the desired part performance. This ensures full utilization of material capability, like previous examples, while also providing robust upstream evaluation. Application of the demonstrated framework has the promise to improve manufacturability and material quality and part performance.

MBMD Workflow Paradigm for Cold-Dwell Fatigue Performance

Another example of an MBMD workflow is in assessing creep-fatigue damage accumulation in titanium rotor components, which has been studied for several decades [49,50,51]. A physics based probabilistic dwell fatigue model framework providing a comprehensive understanding of cold dwell fatigue mechanisms in \(\alpha + \beta\) titanium β alloys was developed and implemented. A series of analytical submodels were incorporated into an explicit Monte Carlo-based probabilistic crack nucleation and propagation model that delineates pre-crack formation and subsequent propagation through hard grains, highlighting the critical role of microstructural features [49]. In this Monte-carlo framework, statistical distributions (M-SERVEs) for microtexture assemblages, shown in Fig. 12, are used to perform numerous iterations, each time using a different set of random samples. In addition, the probabilistic dwell fatigue modeling framework integrates submodels that describe macroscopic and microscopic creep models, dwell-dependent cyclic crack growth models, nucleation criteria, and component fracture criteria, necessary for predicting component life.

Fig. 12
figure 12

Schematic representation of critical MTR assemblage characterized by soft oriented grains (white) close to a hard oriented child grains (red) encompassed by parent (blue) and grandparent (green) hard MTRs

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Another key advantage is the framework’s adaptability to stress-controlled and strain-controlled loading conditions, facilitating simulations for both laboratory specimens and larger stressed components. The incorporation of stress volume in predictive capabilities is a noteworthy feature, allowing for a more accurate understanding of component behavior under different loading conditions. Additionally, the framework effectively models macroscopic stresses and creep, predicting localized strain and cyclic damage within hard-soft oriented grain pairs.

The model incorporates orientation-based microtexture region (MTR) metrics, such as MTR size, density, clustering and spacing. Notably the model accounts for varying degrees of texturing within these regions and quantifies the risk posed by hard-oriented MTRs with low ∆K thresholds. The impact of MTR clustering on crack growth is significant, especially during cyclic crack propagation. The formation and evolution of MTRs result in regions with neighboring features. The hardest regions, termed “child” MTRs, are closest to the original assemblage of α grains during the MTR evolution process, in billet. These child MTRs are often encompassed by “parent” MTRs, which are more diffuse regions of an originally formed MTR. There are also “grandparent” MTRs, which are further diffuse regions of an original MTR region and often contain the child and/or parent MTR regions, as shown schematically in Fig. 12. Another critical element of these MTR assemblages are soft oriented grains, located in the vicinity of hard child MTRs, that accumulate damage and initiate cracks.

Critical zones within a component that are at a higher risk for formation of large MTR assemblages that rapidly accumulate damage are identified via a combination of forge process models, and structural analysis models which consider operational stresses, dwell times and temperatures. Microstructure and microtexture based SERVEs were developed for each of the identified zones or volumes within the rotor by using 2D SEM based EBSD scans (each scan is typically 15 mm × 15 mm) that were subsequently processed in DREAM-3D using specifically developed pipelines that quantify microtexture metrics [49,50,51]. The data from EBSD is segmented and the C-axis misalignment between pixels is calculated to classify pixel groups into child, parent, and grandparent regions. These MTRs are classified based on their hardness and C-axis misalignment. Hardness here is defined as the c-axis alignment with the stress axis, i.e., a hard MTR is well aligned with the stress axis. Clustering metrics of MTRs are calculated using segmentation of the underlying orientation data. Statistically relevant sampling of a material defines how much MTR data must be collected to achieve a specific confidence level. Large, extreme value MTR sizes can influence the MTR statistics for subsequent probabilistic modeling. Therefore, sufficient sampling volume was characterized to ensure these features are adequately included. Figure 13 shows the coefficient of variation for critical number density of collocated child, parent, and grandparent hard MTRs as a function of sampled EBSD area. The dashed vertical line identifies a minimum scan area to achieve a convergence of microtexture features, which were then utilized as input distributions into the dwell fatigue model to predict the probabilistic life to crack initiation and growth of components.

Fig. 13
figure 13

Normalized coefficient of variation (CV) defined as the ratio between standard deviation and the mean, (\({\text{CV}} = \frac{\sigma }{\mu }\)), for grandparent hard MTR size as a function of sampled EBSD area. The dashed vertical line identifies a minimum scan area for converged microtexture features

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Models are also being developed to predict the evolution of key MTR characteristics based on starting billet structures and processing pathways. One such model was worked under an MAI program [6, 50], where compression specimens were extracted from multiple pedigree billets and tested. Location based data sets of strain, strain rate and temperature components were extracted from DEFORM finite element analysis simulations, to determine the key state variables that influence evolution of microtexture from billet to final compressed samples. Final compressed strain, strain rate and temperature variables on location basis were utilized to train a neural network model, which also included average major and minor MTR axis dimensions measured from billet. The neural network was able to generalize the trends between the input and output variables and provide a good predictions of major and minor MTR axis for the validation datasets, Fig. 14.

Fig. 14
figure 14

Neural network model predictions of microtexture aspect ratio (Major axis/minor axis) dimensions after undergoing compression strains typically experienced in component forgings. Blue triangles represent datasets used to train the model and the black circles represent the validation datasets that were not part of the training set. The black dotted line represents a perfect fit bounded by 5% tolerance blue dotted lines. Although, one of the neural network training datasets fell outside the 5% bounds the new validation datasets were accurately predicted

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The analytical models describing various mechanisms within the framework provide a computationally efficient tool for engineering design and structural analysis workflows. Overall, the probabilistic dwell fatigue model offers a sophisticated and versatile approach to understanding and predicting the intricate behavior of materials under cold dwell fatigue conditions. In addition, a framework incorporating PSPP models that predict location based microtexture, properties and component dwell fatigue performance is being developed, as shown in Fig. 15, that will allow the rapid assessment and optimization of processes and rotor designs.

Fig. 15
figure 15

MBMD workflow for microtexture and dwell fatigue performance for rotor components. EBSD images represent MTRs in the millimeter range and are not shown to scale. SERVEs are developed by building up individual scans, each approximately 15 mm × 15 mm in size

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Conclusions

Specific models linked within a PSPP systems-engineering framework have been developed and associated MBMD workflows have been successfully executed in the support of development and qualification of titanium component design and manufacturing processes.

More specifically, the following can be concluded:

  • The MBMD framework enables optimization of process parameters for the enhanced structures (microstructure and microtexture), properties and performance for titanium alloys and components.

  • Convergence of measured microstructure and microtexture distributions enabled the establishment of appropriate SERVEs to probabilistically predict static and dynamic properties in Ti rotors.

  • The MBMD framework enables guided “smart” testing and characterization requirements enabling more efficient and effective qualification and certification of components.