Abstract
This paper develops an uncertaintyquantified parametrically homogenized constitutive model (UQPHCM) for dualphase α/β titanium alloys such as Ti6242S. Their microstructures are characterized by primary αgrains consisting of hcp crystals and transformed βgrains consisting of alternating laths of α (hcp) and β (bcc) phases. The PHCMs bridge lengthscales through explicit microstructural representation in structurescale constitutive models. The forms of equations are chosen to reflect fundamental deformation characteristics such as anisotropy, lengthscale dependent flow stresses, tensioncompression asymmetry, strainrate dependency, and cyclic hardening under reversed loading conditions. Constitutive coefficients are functions of representative aggregated microstructural parameters or RAMPs that represent distributions of crystallographic orientation and morphology. The functional forms are determined by machine learning tools operating on a dataset generated by crystal plasticity FE analysis. For the dual phase alloys, an equivalent PHCM is developed from a weighted averaging rule to obtain the equivalent material response from individual PHCM responses of primary α and transformed β phases. The PHCMs are readily incorporated in FE codes like ABAQUS through userdefined material modeling windows such as UMAT. Significantly reduced number of solution variables in the PHCM simulations compared to micromechanical models, make them several orders of magnitude more efficient, but with comparable accuracy. Bayesian inference along with a Taylorexpansion based uncertainty propagation method is employed to quantify and propagate different uncertainties in PHCM such as model reduction error, data sparsity error and microstructural uncertainty. Numerical examples demonstrate the accuracy of PHCM and the relative importance of different sources of uncertainty.
Introduction
Structural analysis of heterogeneous materials using phenomenological constitutive models, is often faced with inaccuracies stemming from the lack of connection with the material microstructure and underlying physics. Pure micromechanical analysis, on the other hand, is computationally prohibitive on account of the large degrees of freedom needed to represent the entire structure. To overcome these shortcomings, hierarchical multiscale models based on computational homogenization, have been proposed to determine the homogenized material response for heterogeneous materials that can be used in componentscale analysis. A detailed account of these approaches is given in ref. ^{1}. However, for nonlinear problems involving historydependent constitutive relations, many multiscale methods incur prohibitive computational costs from solving the micromechanical problem for every macroscopic point in the computational domain. It is imperative to develop robust macroscale constitutive models, incorporating characteristic microstructural features as well as underlying physical mechanisms of deformation.
The concept of parametric homogenization has been introduced in refs. ^{2,3,4,5} to: (i) overcome prohibitive computational costs of other homogenization methods, and (ii) bridge lengthscales through explicit microstructural representation in structurescale constitutive models. PHCMs for polycrystalline αphase titanium alloys have been developed in refs. ^{1,6,7}. The physicsinformed PHCMs differ from conventional phenomenological models in their unambiguous depiction of constitutive parameters and their dependencies. Coefficients in these thermodynamically consistent, reducedorder constitutive models are functions of representative aggregated microstructural parameters (RAMPs), corresponding to distributions of key morphological and crystallographic descriptors in the microstructure. The forms of equations in PHCMs are chosen to reflect fundamental deformation characteristics of the material, made in micromechanical observations. For the Ti alloys, characteristics like objectivity, anisotropy, tensioncompression asymmetry, and history/pathdependence are represented in the micromechanical crystal plasticity FE (CPFE) models. Constitutive parameters in PHCMs are calibrated from CPFE analysis, and their functional forms in terms of RAMPs are determined using machine learning tools. Details of computational development and validation of the PHCMs for single phase nearα Ti alloys is given in refs. ^{1,6}. The present paper extends the PHCMs to twophase polycrystalline α/β Titanium alloys like Ti6Al4V and Ti6Al2Sn4Zr2Mo that are widely used in aircraft engine components due to their superior mechanical, thermal and fatigue properties^{8}.
Microstructures of dual phase Ti alloys are typically characterized by primary α and transformed β phases, the latter consisting of colonies of alternating laths of secondary α (hcp) and β (bcc) lamallae in a matrix of equiaxed primary α grains as shown in Fig. 1a. The anisotropic phases, colony structures and crystallographic orientation mismatch at grain and lath boundaries significantly influence their deformation, creep, and fatigue characteristics^{9,10,11,12,13,14}. Under dwell loading conditions, the nearα and α/β alloys undergo timedependent load shedding at boundaries between grains with moderate to large lattice misorientation, leading to facet formation signaling the nucleation of fatigue cracks^{15,16,17}. Nucleation and short crack growth in these alloys is strongly influenced by local microstructural features such as crystallography and size of microtextured regions^{9,16,18,19}. These observations call for material microstructurebased modeling to account for morphology and crystallography in the reliable prediction of their mechanical properties and fatigue behavior. CPFE models have been extensively used^{9,13,20} to characterize deformation behavior associated with α and α + β alloys. However, componentscale analysis with these micromechanical models are computationally expensive for macroscopic analysis. The PHCMs in refs. ^{1,6,7} enable fullscale component analysis with explicit representation of microstructural features and characteristics. They enjoy high accuracy, while being many orders of magnitude more efficient than conventional multiscale analysis methods. In this paper, the PHCMs are extended to twophase polycrystalline α/β titanium alloys like Ti6242S, accounting for the presence of the α and β phases in the granular structure. In addition, the PHCM constitutive relations incorporate a kinematic hardening law to account for the Bauschinger effect, an important aspect of the mechanical response under reversed cyclic loading.
In addition, uncertainty quantification and propagation comprise important tasks in the PHCM development process to account for different sources of uncertainties, as depicted in Fig. 2. Uncertainties in constitutive coefficients are attributed to variability in the material microstructure, microstructural characterization, calibration of CPFE model, calibration of PHCM, etc. For example, errors inherently occur in microstructural characterization of electron backscatter diffraction (EBSD) images, used for the generation of microstructurebased statistically equivalent representative volume elements or MSERVEs for polycrystalline microstructures. Generation methods of the MSERVEs for different alloys have been developed in refs. ^{21,22,23}, where the statistical distributions of microstructural descriptors are matched with experimental data obtained from EBSD or scanning electron microscopy (SEM) images. This process also induces uncertainty with respect to matching the higher order statistics of distributions. Consideration of all uncertainties in the multiscale framework is a very detailed process. Three sources of uncertainty in PHCM constitutive coefficients, considered in this paper (see Fig. 2), are:

Model reduction error due to the functional forms used to represent the microstructuredependent constitutive parameters using machine learning tools;

Datasparsity error due to the finite size of the calibration data obtained from a finite number of CPFE simulations used in PHCM calibration;

Uncertainty in the microstructural descriptors or RAMPs due to natural variability in the microstructure.
For predictive analysis of material response, the uncertainties in PHCMs must be adequately quantified and propagated with additional deformation. Bayesian inference, along with maximum likelihood estimation, has been used in ref. ^{14} to quantify (i) the uncertainty associated with constitutive parameters in PHCM functional forms obtained from machine learning, and (ii) the uncertainty due to datasparsity associated with a limited number of microstructures and micromechanical simulations used in calibrating the PHCMs. Furthermore, with evolving deformation, the uncertainty in PHCM constitutive parameters is propagated to response variables like stresses and plastic strains. MonteCarlo based uncertainty propagation is not suitable since it requires repeated evaluations of the PHCMs. Multiple evaluations are generally not desirable for macroscopic analysis under given loading conditions. A Taylorseries expansionbased uncertainty propagation method, originally developed in ref. ^{14}, is used to propagate the uncertainty to response variables. The resulting uncertaintyquantified PHCM has a unique advantage of being incorporated in any commercial software through user material windows without having to perform expensive MonteCarlo simulations.
The paper is organized as follows. The accuracy of PHCM developed in this paper along with the effect of different sources of uncertainty on mechanical response of dualphase titanium alloys is demonstrated through validation examples in “Results” section. The micromechanical crystal plasticity constitutive model and the framework to develop PHCMs for dualphase titanium alloys is given in “Methods” section. Different sources of uncertainty present in the PHCM development are also discussed in this section.
Results
Overview
The PHCM models, developed and calibrated for primary α and transformed β phases in “Methods” section, are used in the volume fraction based equivalent model in Eq. (15) to obtain the equivalent stress–strain response of MSERVEs for a given volume fraction of transformed β grains. The PHCM constitutive equations in “Methods” section are implemented in the user subroutine UMAT of the commercial code Abaqus^{24}. Numerical implementation and integration of the PHCMs in the UMAT have been described in details in ref. ^{6}. In this section, PHCM predictions are compared with micromechanical CPFE simulations under various loading conditions to demonstrate its accuracy and efficiency. This is followed by a numerical example that studies the effect of different uncertainties on the timedependent stochastic mechanical response.
Validating PHCM with micromechanicsbased CPFE simulations
Experimental validation of the PHCMs for a nearα Ti6242S alloy containing ~95% primary α phase has been successfully conducted in ref. ^{6}. However, macroscopic experiments that can be used to validate the PHCMs for dual phase α/β Ti6242S alloys are not available in the open literature, to authors’ knowledge. Therefore, the PHCMs for Ti6242S alloys containing primary α and transformed β phases are validated with micromechanical CPFE simulation results. The microstructure used in the validation process corresponds to the EBSD scan shown in Fig. 3a. It consists of 70% primary α and 30% transformed β grains by volume. The (0001) pole figure of the crystallographic texture for this microstructure is shown in Fig. 3b. The corresponding RAMPs derived from the figures are given in Table 1.
A 503 grain MSERVE is constructed from the EBSD scan of Fig. 3a, for which the (0001) pole figure is depicted in Fig. 3c. This MSERVE is meshed into 492733 linear tetrahedral elements. The PHCM predictions are compared with results of CPFE simulations for four different loading conditions. First, the CPFE model of the MSERVE is subjected to different biaxial stresscontrolled loading conditions. The initial yield stresses along different directions are extracted following the procedure outlined in “Methods” section. The biaxial yield stresses are plotted in Fig. 4a, along with the PHCM yield surface projected on XZ plane. The latter is obtained using anisotropy coefficients calculated from functional forms in Box 1. Excellent agreement is found between the tensile and compressive yield stresses by CPFE simulations and the PHCM. Similarly, uniaxial yield stresses in tension and compression along different directions in each of the XY, XZ, and YZ plane are extracted from results of CPFE and PHCM simulation and compared in Fig. 4b. The difference between the PHCM and CPFE results for the uniaxial yield stresses and their tensioncompression asymmetry is quite small. These comparisons demonstrate the ability of the PHCM to accurately describe material anisotropy, as well as tensioncompression asymmetry.
To assess the accuracy of postyield predictions by PHCM, the MSERVE is subjected to constant strainrate loading along the X and Z directions, both in tension and compression. The homogenized stress–strain response from the CPFE and PHCM simulations is compared in Fig. 4c. The CPFE simulations of the MSERVE take approximately 6 h with 24 CPUs for a true strain of 7%, while PHCMbased ABAQUS simulations of a single element, corresponding to the same microstructure, takes ~1 s on single CPU for the same strain. It is observed that the anisotropy in flow stress is correctly predicted by PHCM both in tension and compression, with a small error in flow stress along the Zdirection under compression.
Finally, the PHCM performance is assessed for cyclic loading by subjecting the MSERVE to straincontrolled, triangular wave and dwell loading as shown in Fig. 5b, d. The triangular cyclic loading profile has a peak strain of ϵ_{0} = 1.2% with a ramp time of 1 s. The dwell loading profile corresponds to a ramp time of 1 s followed by a hold strain ϵ_{0} = 1.2% for 120 s and unloading to zero strain in 1 s. These loadings are chosen to represent typical loading conditions used in fatigue testing of coupon specimens. CPFEMbased dwell simulation of the MSERVE takes approximately 48 h with 24 CPUs, while the corresponding PHCM simulation takes about 3 s on a single CPU for the same number of dwell cycles. Similarly, the triangularwaveform cyclic simulation using CPFEM takes ~35 h with 24 CPUs, while the same simulation using PHCM takes about 3 s on a single CPU. The maximum homogenized tensile and compressive stresses in each triangular cycle by CPFE and PHCM simulations are compared in Fig. 5a for 140 cycles. For dwell loading, the tensile stress at the end of the hold period and the compressive stress at the end of each dwell cycle, obtained by CPFE and PHCM simulations are plotted in Fig. 5c for 90 cycles. The PHCM underestimates the peak tensile stresses and overestimates the peak compressive stresses under both triangular cyclic and dwell loading for this MSERVE. While the error in PHCM prediction is small under dwell loading, it is comparatively high along the Zdirection under triangular cyclic loading. These errors are observed as the applied peak strain lies in a region of the monotonic stressstrain response shown in Fig. 4c, where flow stresses in PHCM do not accurately match those from CPFE simulations. These errors may be attributed to a small mismatch in the elastoplastic transition region during PHCM calibration. This can be improved through incorporating additional functions to better accommodate this transition in PHCM.
Effect of microstructural uncertainty on mechanical response of dualphase Ti alloys
The effect of various sources of uncertainty on the stochastic, timedependent response of the dualphase Ti6242S alloy is studied in this section. The RAMPs for this alloy are given in Table 1.
To study their impact on the accuracy of predictions, the model reduction and data sparsity errors are first considered in stochastic PHCM simulations. These are introduced at the model development and calibration stages of the PHCM. A stochastic PHCM simulation is performed for a uniaxial tension test at a constant strainrate of 1 × 10^{−4} s^{−1} and the corresponding stressstrain response is depicted in Fig. 6a. The black solid line and the gray interval respectively correspond to the mean and the standard deviation of the Cauchy stress. The uncertainty due to model reduction and data sparsity error has a 1σ interval of ~31 MPa corresponding to ~3% of flow stress. This predicted modeling error interval is consistent with the deviation of the PHCM predictions from the CPFE model response, which is shown with markers in the figure. The predicted level of model uncertainty is also consistent with the set of PHCM validation tests in previous section.
Next, the effect of microstructural uncertainty on the mechanical response is analyzed. Microstructural variability, as shown in Fig. 1a, naturally arises during thermomechanical processing of the alloy. The microstructural uncertainty is represented by statistical distributions of RAMPs, which are calculated from spatial variability of the microstructure in EBSD scans using a sampling method that has been discussed in ref. ^{7}. The resulting statistical moments of RAMPs from this sampling method are given in Table 2 and incorporated in the stochastic material evolution by the expansionbased UP method.
To understand the effect of the dualphase α/β microstructure on a representative fatigue problem, the uncertaintyquantified PHCM is used to simulate a fully reversed cyclic loading with a strain amplitude of Δε = 2% and a cycle period of T = 40 s. Only microstructural uncertainty is considered in this analysis. Figure 6b shows the predicted evolution of the mean and 1σ interval of the stressstrain response, arising from microstructural uncertainty represented by the moments of RAMPs in Table 2. At the end of the first cycle, corresponding to the time marked with (*) in the figure, the uncertainty in the stress and plastic strain responses have standard deviation of 23.6 MPa and 1.62 × 10^{−4}, respectively. This represents the variability of the mechanical fields in the material due to crystallographic and morphological variabilities in the microstructure. For example, the fatiguedriving stress at a critical point in a structural component like a bolt or a spot weld, may vary by this amount, depending on the particular microstructure of this critical volume of material that is subject to alleatoric uncertainty.
Multiple simulations are next performed, while individually turning on random variations in different features of the microstructure to identify the important microstructural features that control the uncertainty of material response. Table 3 shows the list of microstructural features and the corresponding stochastic RAMPs used in this analysis. For each class of microstructural uncertainty considered, the table shows the standard deviation of stress and plastic strain response (right) calculated at time (*) marked in Fig. 6b. The relative contribution of each source to the stochastic response is also given in the last column. The two constituents of the alloy, viz., the primaryα grains and the transformedβ colonies, contribute approximately 54% and 46% respectively to the total uncertainty in the response. This is shown in the last row of the Table 3 as "all sources” for each constituent. In the breakdown of microstructural features for the primaryα grains, it is seen that the most important determinant of the stochastic response is crystallography, especially the crystallographic texture (~89%) for this alloy. This dominant effect mainly results from the microtextured regions within the material, i.e., clusters of similarly oriented crystallographic regions. These are visualized in the EBSD scan of Fig. 1a and represented by the moments of \({\overline{OMA}}_{ij}\) in Table 2. Due to anisotropy of the hcp Ti alloy, this variability results in large uncertainty in the stress response. The presence of a high degree of microtexture is associated with poor fatigue performance in Ti alloys^{16,25,26}, due to the production of coherent, continuous slip bands^{27} that induce stressconcentrations at boundaries of the clusters^{28} and propagates shortcracks^{29}.
For the transformedβ colonies, Table 3 shows that two aspects of the microstructure account for most of the uncertainty in the mechanical response. These are (i) the crystallographic texture (58%) due to the presence of microtexture, and (ii) α/β lath thickness (33%). As discussed before, the sizeeffect associated with the α/β interfaces is an important strengthening mechanism in this alloy. The natural variability in the interface thickness results in pockets of high and low strength material, resulting in a variance in the stress and plastic flow. The results indicate that uniformity of the microstructure through reduction of microtexture and having α/β interfaces with similar thickness, can alleviate extreme values of stress and plastic strain in the microstructure. This can improve the fatigue performance of dualphase Titanium alloys.
Discussion
This paper develops an uncertaintyquantified parametrically homogenized constitutive model (UQPHCM) for dualphase α/β Titanium alloys such as Ti6242S. It is an extension of the PHCMs for polycrystalline αphase Titanium alloys developed in refs. ^{1,6,7}. In this paper a class of dualphase α/β Ti alloys containing a single variant of crystallographic orientation corresponding to parallel α/β lamellae in each colony is considered. The authors recognize that this is a limiting assumption, given that colony structures can be quite complex, e.g., in Widmanstätten or "basketweave” microstructures, which are characterized by multiple sets of α lamellae with different variants coexisting within the β grains. This simplifying assumption avoids complex interaction of multiple BOR variants and may affect the polycrystalline deformation mechanisms and response. However, this assumption is deemed sufficient given that the focus of this paper is on the development of an overall multiscale framework.
The micromechanical deformationinformed PHCMs overcome prohibitive computational costs of other homogenization methods, and bridge lengthscales through explicit microstructural representation in structurescale constitutive models. The forms of equations in these thermodynamically consistent models are chosen to reflect fundamental deformation characteristics such as objectivity, anisotropy, tensioncompression asymmetry, and history/pathdependence. Constitutive coefficients are functions of RAMP that represent distributions of key morphological and crystallographic descriptors in the microstructure. The functional forms are determined by machine learning tools operating on a dataset generated by micromechanical analysis. The PHCMs are readily incorporated in FE codes like ABAQUS through userdefined material modeling windows such as UMAT. Significantly reduced number of solution variables in the PHCM simulations compared to micromechanical models, make them several orders of magnitude more efficient, but with comparable accuracy.
An equivalent PHCM is developed to model dualphase polycrystalline α/β Titanium alloys like Ti6Al2Sn4Zr2Mo. The equivalent PHCM assumes a Taylor model of uniform deformation gradient for the two phases, and employs a phase volumefraction based weightedaveraging rule to obtain the equivalent material response from the individual PHCM responses of primary α and transformed β phases. Deterministic PHCMs are developed individually for the primary α and transformed β phases from detailed micromechanical simulations. The equivalent PHCM is consistent with the CPFE model developed in ref. ^{13} and yields accurate results in comparison with CPFEM. A major advantage of this volume fractionbased weighting method over one that couples the effect of both phases into a single unified model is that it avoids the need to create a very highdimensional micromechanical dataset for machine learningbased coefficient function evaluation. This requires a prohibitively large number of CPFE solutions for different MSERVEs by simultaneously varying RAMPs belonging to both phases.
The equivalent PHCMs for dualphase polycrystalline microstructures capture important mechanical behavior observed in Ti alloys, viz., anisotropy, lengthscale dependent flow stresses, tensioncompression asymmetry in initial yield stress and hardening, grain size, lath size and strainrate dependent plastic flow, and cyclic hardening under reversed loading conditions. The resulting constitutive equations consists of an anisotropic elastic model, an anisotropic yield function with tensioncompression asymmetry and an anisotropic hardening model with a nonlinear kinematic hardening rule. The machine learning toolkit^{30} is used to express constitutive parameters as functions of RAMPs that characterize microstructural features such as distributions of crystallographic orientation and misorientation, grain size and lath size. The PHCM performance is assessed by comparing its predictions with CPFE simulations for a validation MSERVE under different loading conditions. The PHCMs are found to yield satisfactory results for all loading conditions. While experimental validation of the PHCMs for a nearα Ti alloy has been successfully conducted in ref. ^{6}, there is a lack of experimental results for validation of the dual phase α/β Ti6242S alloys considered here, in the open literature. Such experimental datasets will comprise results of macroscopic coupon experiments such as tensile tests, along with relevant microstructure characterization data sets. This will be pursued in future studies.
An uncertainty quantification formulation is subsequently developed for the PHCMs to quantify and propagate uncertainties that exist at different stages of its development. Three sources of uncertainty, viz., model reduction error, data sparsity error, and microstructural uncertainty are considered. Consequently, the constitutive coefficients in PHCM are assumed to be random. The probability distributions of these constitutive coefficients are determined using Bayesian inference. The uncertainty in the constitutive coefficients is propagated to response variables such as stresses, plastic strains and state variables using a Taylor series expansionbased uncertainty propagation method. The relative contribution of different sources of uncertainty and the RAMPs towards total uncertainty in stresses and plastic strains is studied through numerical examples.
The uncertaintyquantified PHCMs present a unified framework in which microstructuredependent material response with quantified uncertainties can be efficiently generated. The efficiency and accuracy of PHCMs, along with their easy integration in commercial software, make them highly promising tools for large scale structural analysis.
Methods
Overview
The systematic development of UQPHCM from micromechanical CPFE simulations is described below. Crystal plasticity constitutive model is summarized in the next section for dualphase alloys following which a deterministic PHCM is developed. Different sources of uncertainty in PHCM development are discussed at the end and Bayesian inference is employed to quantify these uncertainties.
Crystal plasticity model for dual phase Titanium alloys
A dualphase Titanium alloy Ti6242S is modeled in this paper. Samples of this alloy have been experimentally characterized in earlier work^{9,13} and exhibits a bimodal (duplex) microstructure as shown in Fig. 1a. The αcolonies in Fig. 1a nucleate from priorβ grain boundaries during cooling of the alloy from the β phase. In general, colony structures can be quite complex such as Widmanstätten or "basketweave” microstructures, which are characterized by multiple sets of α lamellae with different variants coexisting within the β grains^{31,32,33}. These are typically achieved with higher coolingrates or higher content of βstabilizing elements^{34}. The Widmanstätten microstructures involve interaction of a large number of slip systems. Relative crystallographic orientation of colonies belonging to different Burgers orientation relationship (BOR) variants within the transformedbeta phase can introduce complex mechanisms of dislocation transmission or pileup at the colony interfaces. The boundaries between different oriented alpha variants provide strong barrier to dislocation activities, which can affect deformation mechanisms and thereby the mechanical response.
Figure 1a shows different crystallographic orientation variants in the transformed beta phase region. However, there is preponderance of α lamellae in each colony that have identical crystallographic orientations, and thus belong to a single variant of BOR^{32}. With this general observation, the present paper develops a model for a single variant of crystallographic orientation. This simplifying assumption that avoids complex interaction of multiple BOR variants and may have some effect on the polycrystalline deformation mechanisms and response. Since the focus of this paper is on the development of an overall multiscale framework, this simplifying assumption is deemed sufficient. The bimodal microstructure is characterized by equiaxed primary α grains consisting of hcp crystals and transformed β grains consisting of alternating laths of α (hcp) and β (bcc) phases, as shown in the schematic of Fig. 1a.
Depending on the coolingrate, the α colonies may be as large as the prior β grains^{34}, and of comparable size to the primaryα grains in the duplex microstructure^{9}, as shown in Fig. 1a. The volume fraction of the α and β lamellae in the transformed β colonies are assumed to be 88% and 12% respectively, from experimental observations discussed in refs. ^{13,35}. The hcp crystal lattice consists of a total of 30 different slip systems divided into five different slip families, viz., basal \(\left\langle {\bf{a}}\right\rangle (\{0001\}\left\langle 11\overline{2}0\right\rangle )\), prism \(\left\langle {\bf{a}}\right\rangle (\{10\overline{1}0\}\left\langle 11\overline{2}0\right\rangle )\), pyramidal \(\left\langle {\bf{a}}\right\rangle (\{10\overline{1}1\}\left\langle 11\overline{2}0\right\rangle )\), first order pyramidal \(\left\langle {\bf{c}}+{\bf{a}}\right\rangle (\{10\overline{1}1\}\left\langle 11\overline{2}3\right\rangle )\) and second order pyramidal \(\left\langle {\bf{c}}+{\bf{a}}\right\rangle (\{11\overline{2}2\}\left\langle 11\overline{2}3\right\rangle )\). Similarly, the bcc crystal system consists of a total of 48 different slip systems divided into three different slip families—\({{\bf{b}}}_{1}(\{110\}\left\langle 111\right\rangle )\), \({{\bf{b}}}_{2}(\{112\}\left\langle 111\right\rangle )\), \({{\bf{b}}}_{3}(\{123\}\left\langle 111\right\rangle )\). During processing, the nucleation and growth of the α laths within the transformed β matrix follows one of 12 possible crystallographic orientations, given by BOR expressed as (101)_{β}∣∣(0001)_{α}, \({[1\overline{1}\overline{1}]}_{\beta }  {[2\overline{1}\overline{1}0]}_{\alpha }\)^{36}. An example of this relationship is shown in Fig. 1b, which aligns \({{\bf{a}}}_{1}([2\overline{1}\overline{1}0])\) slip direction of hcp crystal with the bccb_{1} (\([1\overline{1}\overline{1}]\)) slip direction. This constitutes a soft slip mode since plastic slip is easily transmitted at the lath boundary^{9}. On the other hand, significant misalignment exists between α phase \({{\bf{a}}}_{2}([\overline{1}2\overline{1}0])\) and β phase b_{2} slip directions, and also between the \({{\bf{a}}}_{3}([\overline{1}\overline{1}20])\) and all \({\left\langle 111\right\rangle }_{\beta }\) directions in the β phase. These slip systems constitute hard slip modes as plastic slip is arrested at the lath boundary. As a result of the soft and hard slip modes, the ease of slip transmission for a_{1}, a_{2}, a_{3} basal and prism slips varies significantly^{13}. Depending on the hard and soft modes of slip transmission between the α and β laths, characteristic lengths are developed for HallPetch type, sizedependent slip system resistances due to dislocation barriers. These developments have been detailed in ref. ^{9} and summarized in the following sections. The colonies are segmented in DREAM.3D software as transformed β grains^{37} and their BOR variants are identified for subsequent MSERVE construction and CPFE analysis.
In the crystal plasticity model, the primary α phase is explicitly modeled using 30 slip systems. A computationally efficient equivalent homogenized model has been developed for transformed β grains^{13}, based on the Taylor model assumption of uniform deformation gradient for all phases with known volume fractions. The model is discussed in the subsequent sections. The equivalent homogenized model has 78 aggregated slip systems corresponding to 30 secondary α slip systems and 48 β slip systems in the transformed β grain. While the α and β lath sizes and shapes can change across the transformed β grains^{38,39}, they are assumed to be constant for all grains in the MSERVE. In addition to the soft and hard slip modes discussed above, the bcc slip systems in {112}〈111〉 and {123}〈111〉 slip families are further divided into soft and hard slip directions. The {110}〈111〉 slip systems exhibit symmetry with respect to the slip direction, while {112}〈111〉 and {123}〈111〉 slip systems are asymmetric with respect to the slip direction^{40}. This leads to a slip directiondependent shear strength for these slip systems, which are considered either soft or hard depending on the slip direction^{13,40}. The crystal plasticity parameters are calibrated using simulations with an imagebased CPFE model and matching the simulation response with those from single crystal and polycrystalline experiments. The general crystal plasticity formulation is the same for both hcp and bcc phases with the only difference introduced in the hardening laws.
Crystal plasticity constitutive relations for hcp and bcc phases
The crystal plasticity model introduces a multiplicative decomposition of the deformation gradient F into a plastic component F^{p} in the stressfree intermediate configuration and an elastic component F^{e} as:
The second Piola–Kirchhoff stress S expressed in the intermediate configuration is related to the elastic GreenLagrange strain E^{e} as:
Here \({\mathbb{C}}\) is a fourth order elastic stiffness tensor that is assumed to represent transversely isotropic and cubic symmetries for the hcp and bcc phases respectively. The plastic velocity gradient L^{p} is related to the plastic sliprate \({\dot{\gamma }}^{\alpha }\) and the Schmid tensors \({{\boldsymbol{s}}}_{0}^{\alpha }={{\bf{m}}}_{0}^{\alpha }\otimes {{\bf{n}}}_{0}^{\alpha }\) for a slip system α with slip normal \({{\bf{m}}}_{0}^{\alpha }\) and slip direction \({{\bf{n}}}_{0}^{\alpha }\), given as:
Here nslip is the number of slip systems (30 for hcp phase and 48 for bcc phase). The plastic sliprate is given by a powerlaw flow rule for both the hcp and bcc phases, as:
where \({\dot{\tilde{\gamma }}}^{\alpha }\) is the reference slip rate, m is the strain rate sensitivity exponent, τ^{α} is the resolved shear stress and χ^{α} is the slip system backstress. Evolution of χ^{α} is given by the ArmstrongFrederick type nonlinear kinematic hardening law^{41} as,
with c and d representing direct hardening and dynamic recovery coefficients respectively. \({\tau }_{{\rm{GP}}}^{\alpha }\) and \({\tau }_{{\rm{GF}}}^{\alpha }\) are the hardening contribution from geometrically necessary dislocations (GND) due to short and long range stresses, given by:
\({c}_{1}^{\alpha }\) and \({c}_{2}^{\alpha }\) are material constants and G^{α}, Q^{α} and b^{α} correspond to shear modulus, activation energy and Burgers vector respectively. \({\rho }_{{\rm{GP}}}^{\alpha }\) and \({\rho }_{{\rm{GF}}}^{\alpha }\) represent GND densities parallel and normal to slip plane and are obtained from the curl of plastic deformation gradient^{17,28}. The slip system hardening resistance g^{α} due to statistically stored dislocations is represented using the following hardening law with stress saturation.
where \({g}_{{\rm{HP}}}^{\alpha }=\frac{{K}^{\alpha }}{\sqrt{{D}^{\alpha }}}\) is the slip system resistance due to the HallPetch effect and D^{α} is the characteristic lengthscale governing the sizeeffect^{9}. D^{α} for different slip systems of hcp and bcc phases are based on the soft or hard slip for a given slip system and are given in Tables 4 and 5 respectively.
The hardeningrate on individual slip systems of the hcp components evolves as,
Here \({h}_{0}^{\beta }\) and \({\tilde{g}}_{{\rm{s}}}^{\beta }\) correspond to the initial hardening rate and saturation stress respectively. r and n are the hardening and saturation exponents respectively. For the bcc phase, the evolution of selfhardening rate is given by,
where \({\gamma }_{{\rm{a}}}=\mathop{\int}\nolimits_{0}^{t}\mathop{\sum }\nolimits_{\beta = 1}^{{\rm{nslip}}} {\dot{\gamma }}^{\beta } \mathrm{d}t\). The parameters \({h}_{0}^{\beta }\) and \({h}_{{\rm{s}}}^{\beta }\) are the initial and asymptotic hardening rates, while \({\tau }_{0}^{\beta }\) and \({\tau }_{{\rm{s}}}^{\beta }\) correspond to initial and asymptotic saturation stresses. γ_{a} is the accumulated plastic slip on all the bcc slip systems at a given point. Finally, a yieldpoint phenomenon is attributed to the resistance due to precipitate shearing \({\tau }_{{\rm{p}}}^{\alpha }\) on the basal and prismatic slip systems in hcp crystals, and is represented using the following relationship^{7}.
Here \({\tilde{\tau }}_{{\rm{p}}}^{\alpha }\) and \({\tilde{\epsilon }}_{{\rm{p}}}^{\alpha }\) are the yieldpoint phenomenon stress and plastic strain magnitudes respectively, and \({\overline{\epsilon }}_{{\rm{p}}}\) is the effective plastic strain. For the bcc phase however, the yieldpoint phenomenon is not seen in experimental observations and is not considered.
It has been observed in experiments that Ti alloys exhibit tensioncompression asymmetry in their overall response. This has been attributed to mechanisms such as residual stresses in colonies during the growth processes and differential slip transmission mechanisms, depending on the loading direction^{9}. The crystal plasticity parameters are separately calibrated in tension and compression and a stressstate dependent, weightedaveraging scheme is employed to determine different slip system parameters. The weight W_{t} for tensile parameters is determined based on the lode angle θ, corresponding to the local stress S state in the πplane. The lode angle is measured clockwise from the third positive principal axis^{42}. The πplane is divided into 6 regions by pure tensile and compressive stress axes. The weighting parameter W_{t} for tensile parameters changes smoothly from 0 to 1 corresponding to pure compressive and tensile stress axes respectively. The weighted averaging ensures that the hardening parameters change smoothly for all the stress states. The weight for tension is given as:
where \({S}_{{\rm{1}}}^{{\rm{dev}}},{S}_{2}^{{\rm{dev}}}\) and \({S}_{{\rm{3}}}^{{\rm{dev}}}\) are the principal values of the deviatoric part of the second PK stress tensor S and J_{2} is its second invariant. A typical hardening parameter, e.g., \({g}_{0}^{\alpha }\) is determined as,
where \({g}_{0}^{\alpha }({\rm{T}})\) and \({g}_{0}^{\alpha }({\rm{C}})\) correspond to initial slip system resistances calibrated from uniaxial tensile and compression experiments.
Equivalent homogenized model for transformed β beta colonies
In CPFE simulations, each grain in the MSERVE is assumed to consist of either primary α or transformed β phase. In transformed β grains consisting of alternating laths of secondary α (hcp) and β (bcc) phases, an equivalent homogenized model has been developed in ref. ^{13} using assumptions of a uniform deformation gradient in the Taylor model for both hcp and bcc phases. The resulting model for colonies consists of an equivalent single crystal with 78 slip systems. The Cauchy stresses σ in each individual phase is:
Using a volumefraction based weightedaveraging rule, the homogenized stress in transformed beta colonies σ^{TB} is obtained from the stresses in individual phases and their volume fractions as,
where \({v}_{{\rm{f}}}^{{\rm{hcp}}}\) and \({v}_{{\rm{f}}}^{{\rm{bcc}}}\) are the volume fractions of the α and β laths in the colonies that are weights. In ref. ^{13}, these are taken to be 88% and 12% respectively.
Calibration of the crystal plasticity model parameters is an important step in the development of UQPHCMs. Deterministic calibration of these parameters from limited number of experimental data sets often results in nonunique parameters. Recently, this uncertainty has been accounted for in refs. ^{43,44} by employing Bayesian calibration, which treats the crystal plasticity parameters as random variables. However, random model parameters necessitates performing stochastic CPFE simulations, and this would significantly increase the computational burden of generating the PHCM calibration dataset. Therefore, deterministic calibration that uses genetic algorithm based optimization has been used to calibrate the crystal plasticity parameters for primary α Ti6242S from single crystal and polycrystalline tension/compression and creep tests in refs. ^{9,13,17,28,45,46}. The calibrated model, along with its validation for the primary α phase are given in ref. ^{1}. Similarly, tension/compression tests for single α − β colonies have been used to calibrate crystal plasticity parameters for secondary α and β phases in refs. ^{9,13}. The CPFE model for the polycrystalline Ti6242S alloy (70% primary α and 30% transformed β phase) has been validated in ref. ^{9} for constant strainrate and creep tests. The calibrated parameters for secondary α and β phases are summarized in Tables 4 and 5 respectively. The resulting calibrated crystal plasticity model is taken to be the reference model, and is used for all simulations in this paper.
Equivalent homogenized model for dual phase polycrystals with Taylor assumption
The idea of the equivalent homogenized model for transformed β colonies given in previous section is extended in this work to model polycrystalline microstructural SERVEs or MSERVEs that contain a combination of primary α and transformed β grains. A typical polycrystalline MSERVEs is illustrated in Fig. 7a, where each grain is either a primary α (white) or a transformed β (black). The volume fraction of transformed β grains is 52% in this MSERVE. The corresponding (0001) and (1120) pole figures of crystallographic texture for primary and secondary hcp phase is shown in Fig. 7b.
To examine the effectiveness of the equivalent homogenized model, the response of the polycrystalline MSERVEs is simulated with three different volume fractions of the transformed β grains, viz., V_{TB} = 22, 52 and 75%. The orientation of the bcc lath in a transformed β grain is determined from the adjacent secondary hcp lath orientation through the BOR. The MSERVE is subjected to a constant, true strainrate loading of 0.001 s^{−1} along the Xdirection. The resulting volumeaveraged Cauchy stresstrue strain plots are shown in Fig. 7c for the three different volume fractions, designated as (CPFE).
For the equivalent model with Taylor assumption, the MSERVE is also simulated by successively assuming only primary α grains (V_{TB} = 0) and transformed β grains (V_{TB} = 1) in the MSERVE. The volumeaveraged stresses in the MSERVE for these two limiting cases are denoted as \({\overline{\sigma }}_{ij}^{{\rm{PA}}}\) and \({\overline{\sigma }}_{ij}^{{\rm{TB}}}\) respectively. Using the volumefraction based weightedaveraging rule with phase volume fractions on the volumeaveraged stresses for each phase, the effective stress in the equivalent model is expressed as:
The corresponding stresses using Eq. (15) are plotted in Fig. 7c and designated as (RoM). The results of the equivalent model match those of the twophase CPFE model accurately for all volume fractions. This approach is consequently used to explicitly represent the effect of the primary α and transformed β phases in the PHCMs of the dualphase Ti alloys discussed next.
Parametrically homogenized constitutive models for dualphase titanium alloys
Physicsbased PHCMs have been developed for polycrystalline microstructures of single phase crystalline materials, e.g., the nearα Ti6242S alloy, in refs. ^{1,6}. The general forms of equations representing the evolution of state variables are chosen to be consistent with microscopic mechanisms of deformation, e.g., anisotropy, tensioncompression asymmetry, hardeningsoftening behavior etc. The first law of thermodynamics is used to bridge lengthscales and express constitutive coefficients as functions of RAMPs. Thermodynamic equivalence, according to the HillMandel condition^{47} defines a homogenized material as being energetically equivalent to a heterogeneous material with polycrystalline microstructures. This paper extends the previous work to dual phase Ti6242S alloys containing primary α and transformed β phases. Steps in the development of the corresponding PHCMs are discussed next.
Sensitivity analysis and identification of RAMPs
The first step in PHCM development is a detailed sensitivity analysis to identify important microstructural descriptors and their distributions that govern the homogenized material response. These microstructural descriptors are characterized by a set of RAMPs, which relate the constitutive parameters in PHCM with the underlying microstructure. Functional dependencies of constitutive parameters in terms of the RAMPs is an important feature of the PHCMs. Different microstructural descriptors and RAMPs that influence the homogenized elastoplastic response of primary α and transformed β MSERVEs are given in Table 6. The associated RAMPs are defined below.
(i) Texture tensor (I^{tex}): The crystallographic caxis orientation distribution of a MSERVE is compactly represented by a texture tensor obtained from the weighted caxes orientations of individual grains. It is defined as:
where \({\hat{{\bf{c}}}}^{(i)}\) and V^{(i)} correspond to the unit vector along the caxis orientation and the volume of i^{th} grain in the MSERVE respectively. n_{G} is the number of grains and V is the total volume of the MSERVE. The eigenvalues g_{α} and eigenvectors v_{α} of the texture tensor I^{tex}, correspond to the texture intensity parameters and material symmetry axes respectively^{6}. This tensor accurately represents the overall elastic stiffness for different crystallographic textures in the polycrystalline ensemble, as demonstrated in next section.
(ii) Lattice orientation with respect to material symmetry axes (OMA): To account for the influence of the orientation of slip systems on the homogenized plastic response, the orientation with respect to material symmetry axes (OMA) is introduced as a RAMP. It is defined in terms of the basal and prism Schmid tensors for each grain with respect to the material symmetry axes, and expressed as:
where \({{\boldsymbol{S}}}_{0,{\rm{basal}}}^{(i)}\) and \({{\boldsymbol{S}}}_{0,{\rm{prism}}}^{(i)}\) are the Schmid tensors for basal and prism 〈a〉slip in i^{th} grain of the polycrystalline ensemble. v_{α} are the unit vectors along the material symmetry axis, derived from the texture tensor I^{tex}. The orientation of the bcc phase in the transformed β colony is determined from the adjacent hcp orientation through the BOR. Consequently, it suffices to characterize the orientation through the OMA functions derived from primary and secondary hcp phase crystallographic orientations. The OMA takes into account the 〈a〉axes distributions of the grains in the microstructure and is able to represent its influence on the overall plastic slip in the MSERVE.
(iii) Grainpair misorientation parameter (\({\overline{A}}_{{\theta }_{{\rm{mis}}}}\)): The effect of misorientation on the homogenized plastic response is represented using a grainpair misorientation RAMP \({\overline{A}}_{{\theta }_{{\rm{mis}}}}\), defined as:
This parameter captures the fraction of grain pairs that have smaller than 15^{∘} misorientation angle θ_{mis} between their \(\hat{{\bf{c}}}\) axes. It is a measure of the ease of slip transfer across grain boundaries.
(iv) Mean and standard deviation of grain size distribution (\({\overline{D}}_{{\rm{g}}}^{\mu }\) and \({\overline{D}}_{{\rm{ln}}}^{\sigma }\)): Size effect in PHCM is represented using mean \({\overline{D}}_{{\rm{g}}}^{\mu }\) and standard deviation \({\overline{D}}_{{\rm{ln}}}^{\sigma }\) of the grain size distribution, which is found to follow a lognormal distribution. The two RAMPs are defined as:
Here D_{i} is the equivalent diameter of i^{th} grain in the MSERVE consisting of n_{G} grains, \({\overline{D}}_{{\rm{g}}}^{\mu }\) is the average grain size and \({\overline{D}}_{{\rm{ln}}}^{\mu }\) and \({\overline{D}}_{{\rm{ln}}}^{\sigma }\) are the mean and standard deviation of the lognormal fit to the grain size distribution.
(v) Alpha and Beta lath thickness (l_{α} and l_{β}): The response of the transformed β phase in the polycrystalline ensemble depends on the distributions of α and β lath thicknesses, in addition to grain size. This is because, the characteristic length in the HallPetch relationship of crystal plasticity model in Eq. (7) depends on the orientation of α and β laths with respect to loading direction. Different characteristic lengths are employed for loading along different directions^{9}. Therefore the thicknesses l_{α} and l_{β} of the α and β laths are considered as RAMPs in the PHCMs for the transformed β phase. However, as the volume fraction of secondary α in colonies is assumed to be constant (88% in the current work), only l_{β} is considered as an independent RAMP. A value of l_{α} ~ 3l_{β} has been determined for 88% volume fraction of secondary α in colonies in ref. ^{9}.
Constitutive equations representing the PHCMs
The general forms of the constitutive equations in PHCMs, representing the homogenized response of polycrystalline microstructures, are chosen to be consistent with microscopic mechanisms of deformation, e.g., anisotropy, tensioncompression asymmetry, cyclic hardening, strainrate dependency etc. Corresponding to the crystal plasticity relations, the homogenized elasticplastic relations for the primary α and transformed β phases are assumed to be similar, with the difference being in the hardening laws. Thermodynamic consistency of these constitutive equations is demonstrated through relations established in the Supplementary Information. The PHCM constitutive parameters are calibrated from homogenized response of CPFE simulations for different microstructures and loading conditions. The sensitivity of the elastic and plastic response on the microstructural RAMPs is delineated in Table 6. The RAMPbased functional forms of constitutive coefficients are discussed next.
RAMPdependent functional forms of anisotropic elastic stiffness tensor
The homogenized elastic stiffness tensor for α phase polycrystalline microstructures has been shown to depend only on the underlying crystallographic texture and single crystal (hcp) elastic stiffness in ref. ^{6}. Similarly, for the transformed β phase, the homogenized elastic stiffness is found to depend on the crystallographic texture, the elastic stiffness of the hcp and bcc phases and their volume fractions. While the hcp and bcc phases are assumed to have transversely isotropic and cubic symmetries respectively, the resulting homogenized elastic stiffness in the transformed β phase may have arbitrary symmetry depending on the crystallographic orientation of the hcp crystals. The bcc laths have a higher stiffness compared to the hcp phase. This leads to an overall increase in elastic stiffness for the transformed β phase over the pure α phase with same crystallographic texture. Orthotropic symmetry has been shown to accurately describe the homogenized elastic stiffness tensor \(\overline{{\mathbb{C}}}\) in ref. ^{6}. The same symmetry is assumed for both primary α and transformed β phase MSERVEs in this study.
The homogenized stiffness tensor \(\overline{{\mathbb{C}}}\) for the polycrystalline MSERVE relates the macroscopic elastic GreenLagrange strain \({\overline{{\boldsymbol{E}}}}^{{\rm{e}}}\) to the macroscopic second PiolaKirchhoff (2nd PK) stress \(\overline{{\boldsymbol{S}}}\) through the relation:
where the macroscopic elastic GreenLagrange strain tensor is defined as:
The macroscopic elastic deformation gradient \({\overline{{\boldsymbol{F}}}}^{{\rm{e}}}\) is obtained from the relation \({\overline{{\boldsymbol{F}}}}^{{\rm{e}}}=\overline{{\boldsymbol{F}}}\ {{\overline{{\boldsymbol{F}}}}^{{\rm{p}}}}^{{\rm{1}}}\), involving the total deformation gradient \(\overline{{\boldsymbol{F}}}\) and the plastic deformation gradient \({\overline{{\boldsymbol{F}}}}^{{\rm{p}}}\). The texture tensor I^{tex} in Eq. (16) is used to express the crystallographic dependency of the elastic stiffness in material symmetry coordinate system \({\overline{{\mathbb{C}}}}_{{\rm{mat}}}\) as:
For primary αphase MSERVEs:
For transformed βphase MSERVEs:
where \({{\mathbb{C}}}_{ijkl}^{{\rm{Eq}}}({{\bf{v}}}_{\alpha })\) is the equivalent single crystal elastic stiffness tensor in transformed β colonies. It is obtained from the single crystal stiffness tensor \({{\mathbb{C}}}_{ijkl}^{{hcp}}({{\bf{v}}}_{\alpha })\) of the hcp phase and \({{\mathbb{C}}}_{ijkl}^{{\rm{bcc}}}({{\bf{v}}}_{\alpha })\) of the bcc phase as:
The nonzero components of the single crystal stiffness tensors from ref. ^{13} are given in Table 7, where v_{1} is the axis of transverse isotropy of the hcp single crystal.
The accuracy of the proposed parametrization in Eqs. (22) and (23) is assessed by comparing its predictions with elastic stiffness coefficients obtained from CPFE simulations on MSERVEs having different crystallographic texture distributions. These distributions are represented by their texture intensity parameters as shown in Fig. 8a. The CPFE simulationbased elastic stiffness components are obtained from the derivatives of the homogenized stresses for prescribed perturbations in each component of the strain tensor^{6}. The accuracy of the PHCM elastic stiffness can be seen from the plots of the normal components of the stiffness tensor in Fig. 8b and c for primary α and transformed β phases respectively. Excellent agreement is found for all components with less than 2% error. Further, it is seen that the transformed β phases have higher elastic stiffness components compared to the primary α phases for a given crystallographic texture.
Consistent with lattice rotation in CPFE model, the material symmetry coordinate system given by the eigenvectors v_{α} of the texture tensor, is assumed to undergo a deformationdependent rotation given as:
R^{mat}(0) corresponds to the initial material symmetry frame constructed from the initial symmetry axes and expressed in a fixed Cartesian coordinate frame with unit basis vectors e_{α} as:
Anisotropic yield function and plastic flow rule in PHCM
The PHCM equations for plasticity must exhibit the following characteristics for consistency with micromechanical observations in CPFEM simulations, discussed in previous sections. They are:

Plastic anisotropy arising from the large difference in slip system resistances for basal/prism and pyramidal slip systems of the α phase.

Tensioncompression asymmetry in the yield surfaces and postyield behavior arising from tensioncompression asymmetry of individual slip systems. Anisotropy and tensioncompression asymmetry is more pronounced in transformed β due to the presence of lath structures with soft and hard slip modes.

Lengthscale dependency due to grain or lath sizedependent slip system resistances expressed in Eq. (7).
The macroscopic plastic deformation gradient \({\overline{{\boldsymbol{F}}}}^{{\rm{p}}}\) is obtained by integrating the macroscopic plastic velocity gradient \({\overline{{\boldsymbol{L}}}}^{{\rm{p}}}\), which is additively decomposed into the rate of deformation \({\overline{{\boldsymbol{D}}}}^{{\rm{p}}}\) and plastic spin \({\overline{{\boldsymbol{W}}}}^{{\rm{p}}}\) tensors. Assuming the plastic spin to be negligible as in crystal plasticity model, the plastic velocity gradient is expressed using the flow rule as:
where \({D}_{0}^{{\rm{p}}}\) and m correspond to the reference strainrate and the ratesensitivity parameter respectively. \(\overline{{\boldsymbol{N}}}\) is the normal to the flow surface. The flow rule in Eq. (27) is assumed to be associative, and the unit vector representing the direction of the rate of plastic deformation tensor is expressed in terms of the normalized gradient of the yield function with respect to the second PK stress as:
where \(\left\Vert \cdot \right\Vert\) is the norm, defined in Eq. (37). Y is the effective transformed stress for an anisotropic yield function^{48} that accounts for tensioncompression asymmetry and is expressed as:
where k is the tensioncompression asymmetry parameter and a is an exponent that governs the smoothness of the yield surface. λ_{1}, λ_{2} and λ_{3} are the principal values of a macroscopic transformed effective stress \(\overline{{{\sum }}}\), given as:
where \({\mathbb{L}}\) is a microstructure dependent, fourth order transformation tensor containing anisotropy coefficients. These coefficients are expressed in the material symmetry coordinate system defined by the eigenvectors v_{α} of the texture tensor I^{tex}. The matrix form of the anisotropic tensor is chosen to be^{49}:
Plastic incompressibility condition leads to the following relation between the components:
Evolution of the back stress \(\overline{{\boldsymbol{\chi }}}\) is governed by a modified ArmstrongFrederick type nonlinear kinematic hardening law given as:
Here C, D, E, G are calibrated constants. The exponential term within brackets manifests a smooth transition from elasticity to plasticity, as observed upon every load reversal in MSERVEs of the transformed β. The expression within the Macaulay brackets is nonzero only during the transition from elasticity to plasticity in every load reversal, when the back stress and the flow direction have opposing directions.
Deciphering constitutive coefficients in terms of RAMPs through machine learning
The dependency of PHCM coefficients on RAMPs like \({\overline{OMA}}_{\alpha \beta }\), \({\overline{D}}_{{\rm{g}}}^{\mu }\) and l_{β} is obtained using machine learning tools. The tools utilize symbolic regression to decipher functional forms of the coefficients in terms of the RAMPs. Unlike traditional regression, where parameters in a given equation are optimized, symbolic regression searches for both the functional form and the coefficients that best fit a given dataset. These functional forms are obtained in this work using a machine learning toolkit Eureqa^{30} that is based on genetic programming^{50}. The code uses calibrated anisotropy coefficients (α_{ii}, γ_{ij}) and the RAMPs (\({\overline{OMA}}_{\alpha \beta }\), \({\overline{D}}_{{\rm{g}}}^{\mu }\) and l_{β}) as inputs. Random functional forms for anisotropy coefficients are generated from different combinations of RAMPs and these are represented as tree structures. Fitness values for each of these equations is evaluated using the input data for anisotropy coefficients and the equations are ranked according to their fitness values. Natural selectionbased genetic operations, such as mutation, cross over and elitism, are used to generate new functional forms for the next generation. This procedure is repeated for a large number of generations until optimum functional forms that best fit the data are obtained. A Pareto front is generated and the final functional form is chosen based on the complexity and accuracy of the functional form.
The anisotropy tensor \({\mathbb{L}}\) in Eq. (31) is expressed in terms of the RAMP \({\overline{OMA}}_{\alpha \beta }\) defined in Eq. (17). This dependencies for MSERVEs of primary α phase have been derived in ref. ^{6} and summarized in Box 1. For MSERVEs of transformed β phase, where soft and hard slip systems strongly govern the plastic response, the transformation tensor \({\mathbb{L}}\) depends on the average grain size \({\overline{D}}_{{\rm{g}}}^{\mu }\) and the β lath size l_{β}, in addition to \({\overline{OMA}}_{\alpha \beta }\). For developing functional forms of the microstructural dependencies using machine learning, a dataset of MSERVEs with different crystallographic orientations, grain and lath sizes is generated. Steps in this process are delineated below.

Three different MSERVEs (M1M3) with different grain size distributions, having mean grain sizes (\({\overline{D}}_{{\rm{g}}}^{\mu }\)) of 3.5, 25.9 and 70.5 μm, are created. Lath sizes l_{α} and l_{β} in these MSERVEs are set to 1 and 0.35 μm respectively. Nineteen different crystallographic textures with different symmetries given in ref. ^{6} are created for the above three sets of MSERVEs to capture the effect of orientation on polycrystalline anisotropy. The combination of orientations and mean grain sizes are chosen such that both soft and hard slip systems are activated, thus giving rise to lengthscale dependent anisotropy. This set of MSERVEs consists of a total of 57 different combinations of grain size and crystallographic texture distributions.

To capture the effect of lath size on plastic anisotropy, the MSERVE M3 is chosen and two different β lath sizes, viz., l_{β} = 0.8 μm and l_{β} = 3 μm are assigned, while maintaining an approximate ratio \(\frac{{l}_{\alpha }}{{l}_{\beta }} \sim 3\). For each of these MSERVEs, 19 different crystallographic textures used above are assigned, creating a total of 38 different MSERVEs.
The anisotropy coefficients α_{ii}, γ_{ij} in Eq. (31), tensioncompression asymmetry parameter k and yield surface exponent a in Eq. (29) are evaluated by calibrating the yield function in Eq. (29) to CPFE simulationbased yield stresses along different directions of the 95 MSERVEs created above. A total of 30 different uniaxial and biaxial stresscontrolled CPFE simulations are performed on each of the above MSERVEs to extract the initial yield stresses^{6} in different directions as shown in Fig. 9. The initial yield stresses are assumed to correspond to an effective plastic strain of 0.3%. Back stresses are assumed to be negligible at this small plastic strain and are not considered while calibrating the anisotropy coefficients. The calibrated anisotropy coefficients for the MSERVE with crystallographic texture shown in Fig. 9a, are given in Table 8 for different grain size distributions and lath sizes. The accuracy of the chosen yield function to describe the anisotropy of MSERVEs of both primary α and transformed β phases is shown in Figs. 9b and Fig. 10a respectively. The values of a and k are calibrated to be −0.1294 and 10 respectively for the transformed β phase. The functional forms of the anisotropy coefficients for MSERVEs of primary α and transformed β are given in Box 1. The indices used in these equations preserve the objectivity of anisotropy coefficients under arbitrary permutations of material symmetry axes labels^{6}. The machine learninggenerated functional forms show interactions between lengthscale parameters such as \({\overline{D}}_{{\rm{g}}}^{\mu }\), l_{β} and orientations represented by \({\overline{OMA}}_{ij}\) in the anisotropy coefficients. Physical insights may thus be gained on the influence of relevant RAMPs on PHCM coefficients representing specific properties.
Anisotropic hardening model in PHCM
Macroscopic anisotropic hardening response of polycrystalline microstructures has its roots in the anisotropy of single crystal slipsystem hardening parameters. Different aspects of the PHCM hardening models for the primary α and transformed β phases, along with the calibration of their microstructural dependency functions, are established in this section.
(i) Hardening laws for the primary α phase: For the primary α phase, the hardening response is characterized by a smooth transition from elasticity to plasticity, followed by a constant hardening slope as shown in Fig. 11a. To account for the transient yieldpoint phenomenon and subsequent hardening, the flow stress Y_{0} in Eq. (27) is represented by a microstructuredependent strain hardening law, which is formulated as a modified Vocetype law^{51} as:
here \({\widetilde{Y}}_{0}\) is the initial flow stress and ψ is a constant, chosen to be 0.75 in this work. \(\hat{\alpha }\) and \(\hat{\beta }\) are microstructure and loading directiondependent parameters that are used to capture the transient yieldpoint phenomenon. \(\overline{H}\) represents the microstructure and loading directiondependent hardening slope. The parameter b controls the rate of cyclic hardening. For Ti alloys, the rate of cyclic hardening is observed to be high for the first ~50 cycles, which is followed by a smaller constantrate. In general, b has been expressed as a function of the size of the plastic strain memory surface^{52}. However in the present work, it is taken to be a calibrated constant for simplicity. The effective plastic strain is defined as:
where \({\overline{{\boldsymbol{D}}}}^{{{\rm{p}}}^{\prime}}={\overline{{\boldsymbol{D}}}}^{{\rm{p}}}\) is the deviatoric part of \({\overline{{\boldsymbol{D}}}}^{{\rm{p}}}\).
To capture the microstructure and loading direction dependency of the hardening parameters, a dimensionless parameter R_{Y} is defined. This parameter accounts for the amount of 〈a〉 type basal and prism slip occuring in the MSERVE for a given loading condition. This parameter has been shown to account for the anisotropy in hardening response of primary α MSERVEs in ref. ^{6}. It is defined as:
Microstructural dependency of R_{Y} in the Eq. (38) comes from the orientation dependency of \({\mathbb{L}}\). The loading direction dependency is accounted for through the second PK stress tensor \(\overline{{\boldsymbol{S}}}\). For the transformed β phase however, R_{Y} also depends on the mean grain size (\({\overline{D}}_{{\rm{g}}}^{\mu }\)) and lath thickness (l_{β}) through \({\mathbb{L}}\). Since for the primary α phase, only the initial slip system resistance is dependent on tensioncompression state, the calibrated tensioncompression parameter k is observed to accurately represent the difference in flow stresses in tension and compression.
(ii) Hardening laws for the transformed β phase: The macroscopic hardening response of the transformed β phase depends on the strainrate and the tensioncompression stressstate, in addition to anisotropy of the single crystal hardening parameters. To capture the ratedependent hardening response, a saturation stressbased hardening law (as in the CPFE model) is assumed for the transformed β phase. The microstructuredependent flow stress is given as:
and the evolution of \({\overline{Y}}_{0}\) is given as:
where \(\overline{H}\) is the microstructuredependent hardeningrate expressed as:
\({\overline{H}}_{0}\) represents the initial hardeningrate and \(\overline{r}\) is an exponent controlling the rate of hardening. Y_{sat} is the saturation stress given as:
where \({\overline{Y}}_{{\rm{sat}}}^{{\rm{Ref}}}\) is a reference saturation stress and \(\overline{n}\) is a saturation exponent, which controls the strainrate dependency of hardening. The PHCM hardening parameters for the transformed β phase are calibrated from CPFE simulations under uniaxial tension and compression loads. The calibration accounts for tensioncompression asymmetry of single crystal hardening parameters for the secondary α phase in the CPFE model. Weighted averaging of the calibrated parameters are used for the PHCM hardening parameters subjected to arbitrary loading. The weighting function W_{t} for tensile hardening parameters is determined from the PHCMbased macroscopic stress tensor using Eq. (11), for consistency with tensioncompression states in CPFEM.
Machine learningbased determination of hardening parameters in terms of RAMPs
As discussed in previous sections, functional forms of the hardening parameters in Eqs. (36), (39), (41) and (42) in terms of RAMPs are determined using the machine learning toolkit Eureqa^{30} that operates on a database created from CPFE simulations of polycrystalline MSERVEs. The microstructural dependency of different hardening parameters in Eqs. (36), (39), (41) and (42) is obtained by calibrating the PHCM coefficients with homogenized response obtained from CPFE simulations of different MSERVEs. The calibration process for primary α has been detailed in ref. ^{6}. Calibration of hardening parameters for the transformed β phase uses the following database of MSERVEs.

Three MSERVEs (M1M3) with different grain size distributions are created. They have mean grainsizes of \({\overline{D}}_{{\rm{g}}}^{\mu }=3.5\ \upmu \mathrm{m},25.9\ \upmu \mathrm{m}\) and 70 μm and standard deviation of 0.11. The α and β lath sizes for these MSERVEs are taken to be 1 and 0.35 μm respectively. Nineteen different crystallographic textures with different symmetries are assigned to the MSERVEs. For a few of the MSERVEs, crystallographic orientations are swapped between grains to obtain three different misorientation distributions. The MSERVEs are subjected to constant strainrate, tensile and compressive loading of 0.001 s^{−1}, along three material symmetry axes as shown in Fig. 11a, b. To calibrate the back stress constants in Eq. (33), straincontrolled cyclic simulations are also performed on these MSERVEs along three material symmetry axes, as shown in Fig. 11a and b. The sinusoidal load profile is applied for 3/4cycle with a peak strain of 3% and a time period of 120 s.

For each of the three grainsize distributions, four different MSERVEs are created with grainsize standard deviations of \({\overline{D}}_{{\rm{ln}}}^{\sigma }=0.12,0.19,0.31,0.39\). Constant strainrate and cyclic simulation tests, discussed above, are performed on these MSERVEs to extract the dependency of hardening parameters on \({\overline{D}}_{{\rm{ln}}}^{\sigma }\).

For MSERVEs of transformed β, lathsize dependency of hardening parameters is determined by simulating the MSERVE M3 with two different lath sizes of l_{β} = 0.8 and 3 μm, having a fixed lathsize ratio \(\frac{{l}_{\alpha }}{{l}_{\beta }}=3\). A few of the 19 different crystallographic textures are assigned to these two MSERVEs. The MSERVEs are simulated under the constant strainrate and cyclic loadings.

To determine the strainrate dependency of transformed β hardening parameters and calibrate the rate sensitivity parameter, a uniform texture MSERVE is created. Constant strainrate simulations for three strainrates, viz., 0.001, 0.01 and 0.1 s^{−1} are conducted in both tension and compression.

For the MSERVE M1 with 19 different crystallographic textures, straincontrolled dwell simulations are conducted along the three material symmetry axes to comprehend stress relaxation behavior with cycles. The imposed dwell loading consists of a hold strain of ϵ_{0} = 1.2% with a ramp time of 1 s and a hold time of 120 s, as shown in Fig. 5d. The tensile stress at the end of strain hold and the compressive stress at the end of each cycle are obtained from CPFE simulations. The PHCM calibration process for both the primary α and transformed β phases are plotted in Fig. 11c and d respectively. Since stress relaxation is observed to occur predominantly in the first few cycles of loading, only 10 dwell cycles are simulated for the MSERVEs.
The homogenized stressstrain responses from the constant strainrate and cyclic load simulations of all the MSERVEs constitute a representative database. These datasets are used by the machine learning toolkit^{30} for calibrating the hardening parameters in tension and compression. The resulting functional forms of the hardening parameters in terms of RAMPs are given in Box 2.
Uncertainty quantification (UQ) and uncertainty propagation (UP) in PHCM
Uncertainty in the multiscale PHCMs may be derived from multiple sources. These may be broadly represented by the following considerations:

Model reduction error: The functional forms of constitutive parameters in the PHCMs, e.g., those expressed in Box 1 and Box 2, are in essence reduced order models of the true microstructuredependency that is implicit in the micromechanical model. A certain level of model reduction error inherently exists in the machine learninggenerated constitutive parameters, which propagates to the predicted material response variables. e.g. the macroscopic Cauchy stress \(\overline{{\boldsymbol{\sigma }}}\left(t\right)\).

Data sparsity error: A source of uncertainty in the PHCM response is due to the finite size of the calibration dataset. For example, the number of MSERVEs included in the PHCM calibration dataset is limited by the computational cost of performing the CPFE analyses. This collection of MSERVEs cover a finite domain in the RAMPspace, and predictions made for microstructures outside this domain contain additional uncertainty as a function of distance from the calibration data points. This uncertain error is referred to as the data sparsity error, and is represented by the Bayesian posterior distribution of model coefficients. Within a Bayesian calibration framework, it may be shown that this type of uncertainty depends on both the number and distribution of the calibration data points in the RAMPspace^{14}. For a given microstructure, the total deviation of the PHCMpredicted response from the micromechanicsbased response is accounted for by the model reduction and data sparsity errors.

Microstructural uncertainty: Microstructural uncertainty may be due to limited experimental characterization data or due to inherent aleatoric uncertainty of the microstructure, which naturally arises from the manufacturing process. This uncertainty necessitates RAMPs to be represented as stochastic variables, rather than deterministic quantities.
During the development of PHCMs, the uncertainties due to model reduction and data sparsity are assessed by computational validation studies following model calibration, by comparing PHCM predictions with CPFEM results. For the dualphase Ti alloys, these comparisons have been performed in "Results" section. However, the validation studies are performed only once, with a certain set of microstructures and load cases. In general, the prediction error will be unknown to the enduser for the particular microstructure and load case of interest. In addition, quantification of the effect of the microstructural uncertainty, in the model response by conventional samplingbased methods like the Monte Carlo methods requires repeated model evaluations. This is computationally prohibitive for any practical application. It is therefore desirable to have a builtin capability to evaluate the uncertainty in PHCM response, on the fly, as the analysis is taking place within a commercial FE software.
To address these needs, stochastic PHCMs with uncertainty quantification and builtin uncertainty propagation capabilities are developed in this paper for dualphase Titanium alloys. The methods follow a framework that has been introduced in ref. ^{14}. Assuming the CPFEM simulationbased micromechanical response as the true model, the three main sources of uncertainty discussed above are considered in this framework. A Taylorexpansionbased uncertainty propagation (UP) method calculates the uncertainty in the time and historydependent material response variables like the macroscopic Cauchy stress, plastic strain and fatigue indicators. A Bayesian framework that is used for the calibration of UQenhanced stochastic PHCMs is summarized below, along with the formulation of the builtin UP method.
Formulation of stochastic PHCMs with Bayesian inference
In the stochastic PHCMs, functional forms of the microstructuredependent constitutive parameters are extended to probabilistic models as:
where \({\boldsymbol{\theta }}=\left\{{\theta }_{i}\right\}\) are the microstructuredependent PHCM constitutive parameters identified in equations given in Boxes 1 and 2. \({\boldsymbol{\phi }}\left({{\bf{x}}}_{{\rm{R}}}\right)=\left\{{\phi }_{k}\left({{\bf{x}}}_{{\rm{R}}}\right)\right\}\), k = 1, . . ., n_{b} are the basis functions identified by machine learning in terms of the microstructurebased RAMPs x_{R}, and \({\bf{C}}=\left[{C}_{ik}\right]\) are random coefficients identified by Bayesian inference from CPFEbased calibration dataset. \({{\boldsymbol{\varepsilon }}}_{{\rm{m}}}=\left\{{\varepsilon }_{{\rm{m}},i}\right\}\) are terms in the model reduction error, which model the disparity between the unknown true values of the constitutive parameters for a given microstructure and the prediction of the functional form. These terms are assumed to be independent Gaussian random variables with zero mean and variances \({\sigma }_{{\rm{m}},{\rm{i}}}^{2}\), i.e.,
The set of hyperparameters σ_{m,i} are identified initially by the maximum likelihood estimation, and held as constant during the Bayesian inference of coefficients C, discussed next.
The CPFEgenerated dataset for PHCM calibration, denoted by \({\mathcal{D}}\), is used for the Bayesian inference of the stochastic PHCMs. The Bayesian update for the PHCM coefficients C is expressed as:
where \(p\left({\bf{C}}\right)\) and \(p\left({\bf{C}} {\mathcal{D}}\right)\) are respectively the prior and posterior distributions of the coefficients. \(p\left({\mathcal{D}} {\bf{C}}\right)\) is the joint probability of producing the data \({\mathcal{D}}\), given the PHCM coefficients C, which forms the likelihood function \({\mathcal{L}}\left({\bf{C}};{\mathcal{D}}\right)=p\left({\mathcal{D}} {\bf{C}}\right)\) on the parameter space of C. Each evaluation of \({\mathcal{L}}\) involves simulations of the input data with stochastic PHCM, i.e., the collection of MSERVEs under load cases described in previous section. Noninformative, flat priors \(p\left({\bf{C}}\right)\propto 1\) are used to avoid subjective bias of the posterior distribution. The posterior distribution \(p\left({\bf{C}} {\mathcal{D}}\right)\) is sampled using the Hamiltonian Monte Carlo technique^{53} with NoUTurn sampler^{54}. The uncertainty due to calibration data sparsity is entirely contained in the Bayesian posterior distribution \(p\left({\bf{C}} {\mathcal{D}}\right)\). The statistical moments of the posterior are calculated from the collected samples and are embedded in the stochastic PHCMs for macroscopic response predictions.
The above framework is used for Bayesian inference of the functional forms of the yield surface transformation parameters \({{\boldsymbol{\theta }}}_{{\rm{YF}}}=\left\{{\alpha }_{ii},{\gamma }_{ij}\right\}\) of the matrix \({\mathbb{L}}\) in Eq. (31). The yield function calibration dataset \({{\mathcal{D}}}_{YF}\) is generated in the previous section for (a) αphase MSERVE and (b) transformedβ phase MSERVE. Figure 12 shows the calibrated yield function for two of the calibration microstructures (lines) compared to their CPFEMbased yield points collected under biaxial tests (markers). The black solid line is the mean yield surface and the gray shaded region is the 1σ interval corresponding to the model reduction error. The CPFEMbased yield points from the calibration dataset \({{\mathcal{D}}}_{{\rm{YF}}}\) (markers) are used to identify the hyperparameters \({\sigma }_{{\rm{m}},{\rm{i}}}^{2}\) of the model reduction error in Eq. (46), as well as to perform Bayesian inference of the form coefficients based on Eq. (48). Figure 13 shows yield surfaces predicted by the Bayesian posterior distribution for two experimentally characterized microstructures, which are not included in the calibration dataset \({{\mathcal{D}}}_{{\rm{YF}}}\). The black solid line is the posterior mean yield surface and the 1σ interval bounded by the dashed lines corresponds to the total uncertainty in the posterior prediction, including both model reduction and data sparsity errors. The darkgray shaded region is the 1σ interval corresponding to the model reduction error only. The probabilistic predictions are compared to the CPFEMbased yield points (black markers). Figure 13c shows zoomedin views of these comparisons.
It is emphasized that in contrast to deterministic PHCMs, the material constitutive parameters θ in stochastic PHCMs, e.g. yield function parameters \(\{{\alpha }_{ii},{\gamma }_{ij}\}\), hardening parameters \(\{{\widetilde{Y}}_{{\rm{0}}},\hat{\alpha },\hat{\beta },\overline{H}\}\), etc. are random variables. This is due to the randomness of the form coefficients C, the model reduction error terms ε_{m}, and the RAMPs x_{R}. Consequently, all material state and response variables like \({\bar{\varepsilon }}_{{\rm{p}}}\), \({\overline{{\boldsymbol{F}}}}^{{\rm{p}}}\), \(\bar{\boldsymbol{\sigma} }\) etc. become correlated random variables, whose joint probability distribution evolves with deformation. A method for propagation of the uncertainty through material rate equations is presented next.
Builtin uncertainty propagation in stochastic PHCMs
A Taylor series expansionbased uncertainty propagation (UP) method has been developed in ref. ^{14} to propagate uncertainty among the PHCM constitutive parameters and evolving material state and output variables. This method is extended to PHCMs for the dualphase Ti alloys in this paper. First, the operation of the deterministic material timeintegration algorithm of PHCM is expressed as:
where \({\boldsymbol{\xi }}=\left\{{\xi }^{i}\right\}\) is an evolving state vector. It contains all microstructuredependent constitutive parameters \({\boldsymbol{\theta }}=\left\{{\theta }_{i}\right\}\), as well as the material state variables like the plastic deformation gradient \({\overline{{\bf{F}}}}^{p}\), backstress \(\overline{{\boldsymbol{\chi }}}\) and effective plastic strain \({\bar{\varepsilon }}^{p}\). The state vector ξ may be generalized to a stochastic vector with a timedependent joint probability distribution \(p\left({\boldsymbol{\xi }};t\right)\). The mean and covariance of the stochastic state vector ξ are respectively defined as:
for the time step t_{(n)}. The nonlinear operator \({\mathcal{I}}\left[{{\boldsymbol{\xi }}}_{(n)}\right]\) is expanded in the neighborhood of ξ_{(n)} = μ_{(n)}, which allows one to calculate the moments (μ_{(n+1)}, ∑_{(n+1)}) at time t_{(n+1)} in terms of (μ_{(n)}, ∑_{(n)}) at time t_{(n)}. The resulting timemarching rules are expressed using Einstein index notation as:
where \(\{{\overline{{\mathcal{I}}}}^{i}\}={\mathcal{I}}[{{\boldsymbol{\mu }}}_{(n)}]\) is the timeintegrator evaluated with the mean state μ_{(n)} at time t_{(n)}. The quantities \({{\mathcal{I}}}_{k}^{i}\), \({{\mathcal{I}}}_{km}^{i}\) etc. are partial derivatives of the timeintegrator, i.e.,
evaluated at ξ_{(n} = μ_{(n)}. The fourth order central moment \({\Sigma }_{(n)}^{kmrs}\) is approximated using Isserlis’ theorem^{55}, based on \({\Sigma }_{(n)}^{km}\). In the limit \({\Sigma }_{\left(0\right)}^{km}=0\), this scheme is equivalent to deterministic timeintegration with μ_{(n)} = ξ_{(n)}, Σ_{(n)} = 0 for all t_{(n)}. Similar expressions, based on series expansions are also derived to propagate the uncertainty from the material state variables to output variables, e.g., Cauchy stress, at every time step. The builtin UP method has been verified by comparison of the calculated stochastic response with Monte Carlo simulation results in ref. ^{14}.
Data availability
The datasets generated during and/or analyzed during the current study will be available from the corresponding author on reasonable request.
Code availability
The codes that are used to generate the results in the current study will be available from the corresponding author in a suitable format upon reasonable request.
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Acknowledgements
This work has been partially supported through a subcontract to JHU (subrecipient) from the Ohio State University through a subaward No. 60038238 from an AFRL grant No. FA86501322347 as a part of the AFRL Collaborative Center of Structural Sciences. The work has also been supported by the Air Force Office of Scientific Research Structural Mechanics and Prognosis Program through Grant No. FART1645 (Program Manager: Dr. J. Tiley). Computing support from Hopkins High Performance Computing Center (HHPC) and Maryland Advanced Research Computing Center (MARCC) is gratefully acknowledged.
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S.G. has conceived the overall idea and contributed to the development of the PHCM and CPFE models. He has also revised and wrote parts of the manuscript. Both S.K. and D.O. have developed the models and codes for the deterministic PHCM, produced the results and written the manuscript. D.O. has developed the stochastic PHCM framework.
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Kotha, S., Ozturk, D. & Ghosh, S. Uncertaintyquantified parametrically homogenized constitutive models (UQPHCMs) for dualphase α/β titanium alloys. npj Comput Mater 6, 117 (2020). https://doi.org/10.1038/s41524020003793
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