Abstract
The optimal design of shape memory alloys (SMAs) with specific properties is crucial for the innovative application in advanced technologies. Herein, inspired by the recently proposed design concept of concentration modulation, we explore martensitic transformation (MT) in and design the mechanical properties of TiNb nanocomposites by combining highthroughput phasefield simulations and machine learning (ML) approaches. Systematic phasefield simulations generate data of the mechanical properties for various nanocomposites constructed by four macroscopic degrees of freedom. An MLassisted strategy is adopted to perform multiobjective optimization of the mechanical properties, through which promising nanocomposite configurations are prescreened for the next set of phasefield simulations. The MLguided simulations discover an optimized nanocomposite, composed of Nbrich matrix and Nblean nanofillers, that exhibits a combination of mechanical properties, including ultralow modulus, linear superelasticity, and nearhysteresisfree in a loadingunloading cycle. The exceptional mechanical properties in the nanocomposite originate from optimized continuous MT rather than a sharp firstorder transition, which is common in typical SMAs. This work demonstrates the great potential of MLguided phasefield simulations in the design of advanced materials with extraordinary properties.
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Introduction
Titaniumbased shape memory alloys (SMAs), such as TiNb alloys, are an important class of smart materials that possess shape memory effect (SME) and pseudoelasticity (PE)^{1}, as well as high specific strength, excellent corrosion resistance, superior biocompatibility^{2,3}, etc. These fascinating merits are widely exploited for industrial^{4} and biomedicine applications^{5}. In general, the SME and PE originate from temperature or/and stressinduced reversible firstorder martensitic transformation (MT)^{6,7,8}. The common PE stressstrain curve during loading consists of an initial true elasticity stage with a high Young’s modulus (>80 GPa) of austenite phase until MT occurring, following by the stressplateau associated with the structural transition, and final true elastic deformation of martensitic phase, while during unloading the stress lever causing the reverse MT is lower than that causing MT during loading, thereby exhibiting a large stressstrain hysteresis. Due to the stressstrain hysteresis, low efficiency and poor position control of the SMA actuators are generally observed due to the large hysteresis and strong nonlinearity^{9}. Moreover, as potential metallic biomaterials, a low Young’s modulus comparable with those of natural human bones (~20 GPa^{10}) is essential for TiNb alloys to avoid the “stressshielding effect”^{11} and the resulting bone degradation. Although several efforts have been made to optimize the mechanical responses of metallic alloys, e.g., modulation of the components and concentrations^{12,13}, defect engineering^{14,15}, introduction of elasticinelastic strain matching^{16,17}, and grain refinement^{18,19}, the deliberate design and control of MTs for a combination of specific properties, such as low modulus, linear superelasticity, and freehysteresis is still highly desired for various advanced biomedical and engineering applications.
Recently, Zhu et al.^{20,21,22,23,24}. have conducted comprehensive investigations via phasefield simulations on the design of how to make apparently linearsuperelastic alloys with ultralow modulus and high strength in SMAs. The pioneer works illustrate that concentration modulation, which can be obtained by spinodal decomposition^{20,21,22}, multilayer deposition^{23}, and precipitation of nanoprecipitates^{24}, is able to convert the firstorder MT to pseudohighorder MT and result in purely apparent elastic deformation with low Young’s modulus and high strength. From the perspective of the Landau thermodynamic theory, such manipulation of the mechanic properties essentially derives from the flattening the average thermodynamic energy density profile of the structures, since many physical quantities of ferroic materials are associated with freeenergy derivatives with respect to certain thermodynamic variables. More specifically, elastic constants of ferroelastics such as SMAs are determined by second derivatives of the Gibbs freeenergy density with respect to strain^{25,26}, i.e., the curvature of the thermodynamic energy profile. This design concept has also been widely used in ferroelectrics to optimize corresponding properties, such as ultrahigh piezoelectricity^{27,28} and ultrahigh energy density dielectrics^{29} achieved by deliberately introducing nanoscale structural heterogeneity or nanodomain.
On the other hand, the rational design of these concentration modulations or nanoscale structural heterogeneity usually requires synergistic control of various structural attributes (such as concentration gradient, distribution, and morphology, etc) and chemical features, which poses a great challenge for the conventional timeconsuming experimental or computational searches by the trialanderror process. The emerging datadriven material design approach through the combination of highthroughput (HTP) calculations and machine learning (ML) has been recognized as a powerful tool in the search for new materials or structures with tailored properties and novel functionalities^{30}. There have been many inspirational HTP firstprinciples works, from which direct links between atomicscale information and macroscopic functionalities are established. These approaches have shown many successes in the discovery of new piezoelectric and dielectric materials^{31}, thermoelectric materials^{32}, magnetic materials^{33}, and so on. Nevertheless, the origin of material properties resides in not only chemical constituent itself but also mesoscale morphological and microstructure evolutions^{34}. Mesoscale phasefield simulations in a highthroughput manner that allow for the investigation of microstructure effects are rather limited^{35,36,37}. The development of HTP mesoscale calculations is not only indispensable for the accelerated characterization of microstructure evolutions but also a boost for data and modelingdriven discovery of new materials and structures.
In this work, we propose an alternative concentration modulation approach, i.e., nanocomposite engineering, meanwhile developing an integrated framework of HTP phasefield simulations and ML, to achieve the extraordinary mechanical properties in the TiNb SMAs. A HTP phasefield simulations procedure is designed to optimize the mechanical response of various TiNb SMA nanocomposites constructed by four structural feature variables and to seek the optimal microstructure with desired properties. Guided by the ML, a nanocomposite with a perfect combination of ultralow modulus, quasilinear elasticity, and nearzero hysteresis is screened out, which shows great potential for biomedical material applications. The developed computational approach is also applicable to the design of a wide range of advanced materials.
Results
Workflow of highthroughput phasefield simulations
We design a series of TiNb squarearray nanocomposite structures^{38} consisting of different Nblean nanofillers that are uniformly and coherently embedded in a Nbrich TiNb alloy matrix (Fig. 1b). In the simulations of uniaxial tension tests, the displacement of the lowerleft corner and the xdirection displacement of the left side of the nanocomposite are fixed at zero to prevent the rigid body translation and rotation (see Fig. 1b). The motivation for the choice of such a nanostructure is the advancements in fabrication techniques of composites and their various associated phenomena^{39,40,41}. The TiNb nanocomposite has four structural feature variables (macroscopic degrees of freedom) related to the macroscopic mechanical properties, namely the Nb concentration (C_{Nb}) of the matrix (M_{Nb}) and nanofillers (F_{Nb}), volume fraction of nanofillers (V_{F}), and the number of nanofillers (N_{F}). Following the novel approach of concentration modulation and concentration gradient layer structure developed by Zhu et al.^{20,21,22,23}, M_{Nb} and F_{Nb} are set with 15–20 at.% and 5–10 at.%, respectively, with a 1% interval, to facilitate the stressinduced MT during loading and inverse MT during unloading. To systematically study the effect of V_{F}, a series of nanofillers with the total areas of 8^{2} nm^{2}, 16^{2} nm^{2}, 24^{2} nm^{2}, 32^{2} nm^{2}, and 40^{2} nm^{2} are adopted in the simulations, which correspond to the variation of V_{F} from 1.56% to 39.06%. Correspondingly, three groups of N_{F} are defined: 4, 16, and 64. The detail of four structural feature variables are also summarized in Supplementary Table 1 in the Supplementary information. The computational framework for the design of the TiNb nanocomposite is represented schematically in Fig. 1c. A computing script framework is developed to concurrently handle multiple calculation tasks. With a multicore workstation, more than 10 tasks are concurrently calculated. We start by specifying the above four features (M_{Nb}, F_{Nb}, V_{F}, and N_{F}) with automatic nanocomposite modeling, which yields 540 candidates from a combinatorial point of view. Although the nanofillers could be randomly distributed in the matrix, we restrict the nanofillers to a square nanocomposite structure^{38}, a typical Archimedean lattice structure, to reduce the computational complexity. We then carry out HTP phasefield simulations for the established nanocomposite models and compute their microstructure evolutions and stressstrain (SS) curves under mechanical stress along the [100] direction as shown in Fig. 1b. The quantitative analyses of the SS curves are subsequently conducted, and the main mechanical properties are extracted, including the apparent incipient Young’s modulus (E_{I}), the apparent elastic stress limit (σ_{L}), and the hysteresis area (A_{H}). In the present study, the apparent elastic limit is defined as the critical point of the apparent elastic stage on a stressstrain curve obtained by the “backward convex splitting method”, as schematically explained in Figure S7 in the Supplementary information, where ε_{L} and σ_{L} denote the apparent elastic strain limit and associated stress, respectively. The apparent Young’s modulus is defined by E_{I}=σ_{L}/ε_{L}. The hysteresis area of A_{H} measures the size of the hysteresis area encircled by the whole SS curve. Based on the data sets from HTP phasefield simulations, we employ ML strategy to perform multiobjective optimization of the mechanical properties for the nanocomposites. The recommended candidates are further verified by the phasefield simulations and the underlying mechanisms are also clarified.
Overview of highthroughput phasefield results
Based on the HTP phasefield simulations, we plot the obtained four features dependence of the mechanical properties of the nanocomposites in Fig. 2, in which V_{F} increases from 1.56 to 39.06% on each row and N_{F} decreases from 64 to 4 on each column. It can be seen that E_{I} and ε_{L} of the nanocomposites are widely distributed within 10–35 GPa and 0–3%, respectively. This tunable and variable modulus indicates that engineering the chemical composition and geometry of the nanocomposite offers great flexibility to tune the mechanical properties of the materials. According to the map shown in Fig. 2a, there is a rough trend that nanocomposites with larger V_{F} exhibit lower E_{I}, although the Nblean nanofillers are assumed to have the same elastic constants as that of the matrix. It is also interesting that F_{Nb} dramatically reduces E_{I} at a lower F_{Nb} (F_{Nb} < 8%) (see Fig. 2a) and higher V_{F} (V_{F} > 14.06%), while M_{Nb} and N_{F} are negligible on E_{I} for the whole V_{F} range. These results suggest that coupling between nanofillers and matrix with different C_{Nb} gives rise to E_{I} superior to their inherent ones. As displayed in Fig. 2b, σ_{L} also decreases with increasing V_{F}, which is consistent with the conclusion of E_{I}. In comparison, σ_{L} increases with increasing M_{Nb}, indicating that the Nbrich matrix effectively maintains the linear elasticity of the nanocomposites. The opposite variation tendencies are observed for E_{I} and σ_{L} with F_{Nb} and M_{Nb}, manifesting that F_{Nb} and M_{Nb} play different roles in regulating the mechanical properties of the nanocomposites. Moreover, the calculated A_{H} of the nanocomposites spans a large range from 0 MJ m^{−3} to 3 MJ m^{−3} for different features (Fig. 2c), and thus the embedding of nanofillers greatly reduces the energy loss during the loadingunloading cycle. Although the regulated effect of N_{F} is not as obvious as other features, it can be seen from Fig. 2a to c that N_{F} is also of great significance in finetuning mechanical properties of nanocomposites. All these results indicate that the mechanical properties of nanocomposites are nonlinearly affected by all four features.
Effect of the coherent nanocomposite
The widely tunable and variable mechanical responses of the nanocomposites could be attributed to the change in the MT nucleation critical stress and the local stress fields associated with the inserted nanofillers. Figure 3a plots the Landau freeenergy density curves as a function of the order parameter for the TiNb alloys with different C_{Nb} at 300 K, in which the martensitic phase gradually transformed from an unstable state to a stable state with decreasing C_{Nb}. It is evident that the martensitic phase is stable in the Nblean cases (C_{Nb} < 8%), while in the Nbrich cases (C_{Nb} ≥ 8%), the martensitic phase could not nucleate without the aid of external loads. The critical stress for the stressinduced martensitic nucleation versus C_{Nb} is further calculated (see Fig. 3b) and the results demonstrate that the critical stress increases exponentially with increasing C_{Nb}. For the heterogeneous nanocomposites, martensite thus preferentially nucleates in Nblean nanofillers and generates local stress fields associated with the C_{Nb}dependent stressfree transformation strain (SFTS) tensors (see Fig. 3b) to promote the growth of martensite into the matrix, while the high critical stress in the Nbrich matrix will confine the extent of growth. The critical stress for the nucleation (or growth) of martensite is the minimum stress for the nucleation (or growth) of martensite transformed from austenite in a homogeneous system. To qualitatively investigate and clarify the mechanism for the modulation of mechanical properties, we adopt a simplified 2dimensional model with the nonperiodic condition and stressfree condition to explore the competitive relationship between the Landau energy and the elastic energy. It consists of 8 nm^{2} martensite particle with Nb concentration of f_{Nb} (denoted as the green square) and the adjacent 8 nm^{2} austenitic matrix with Nb concentration of m_{Nb} (denoted as the orange square), corresponding to the martensite volume fraction of 0.5. After relaxation, the interface between martensite and austenite moves toward the particle or matrix depending on f_{Nb} and m_{Nb}. Different combinations of f_{Nb} and m_{Nb} will generate various martensite volume fractions as shown in Fig. 3c. The heatmap of martensite volume fraction after relaxation is divided by a dashed line into the suppressed and nonsuppressed region, which is highlighted in blue and red colors, respectively. These results thus indicate that F_{Nb} (f_{Nb}) and M_{Nb} (m_{Nb}) significantly mediate the Landau freeenergy and SFTS, and further regulate the MT nucleation critical stress and spread of martensite in the nanocomposites.
Furthermore, the geometric features (i.e., N_{F} and V_{F}) modulate the strength and distribution of the generated stress field. We perform simplified simulations to investigate the local stress field σ_{11} (Fig. 3d–r) generated by the lattice mismatch between the embedded martensitic fillers (V3 with F_{Nb} = 7%) and the matrix with different N_{F} and V_{F}. The local stress field probability distribution is further summarized in Fig. 3su. We observe that an increase in V_{F} resulted in the gradual expansion of the local stress distribution range, such that V_{F} greater than 25% produces a second stress distribution peak around 400 MPa. In addition, the increase of V_{F} not only increases the maximum value of the local stress field (represented by the red dashed line) but also significantly increases the probability that the local stress is greater than the critical stress of martensite spread into the matrix, as indicated by the blue dashed line in Fig. 3s–u for the median value (i.e., 600 MPa). Moreover, N_{F} also regulates the distribution and probability density of the local stress field. Contrary to V_{F}, a smaller N_{F} increases the maximum value of the local stress field and the probability of σ_{11} ≥ 600 MPa. Hence, V_{F} and N_{F} effectively affect the magnitude and extent of the local stress field. The shear component σ_{12} also contributes to the nucleation and growth of the MT process. The high shear stress region in the nanocomposite beyond the critical shear stress is very small and highly localized, as shown in Figure S5 in the Supplementary information. Therefore, all four features regulate the nucleation and spread of the MT process by altering the MT nucleation critical stress and the local stress field distribution.
Machine learning guided optimization of the nanocomposites
As previously discussed, reducing the Young’s modulus while maintaining a large ε_{L} is critical for the ideal orthopedic implant. Through the HTP phasefield simulations of the initial hundreds of nanocomposites, we have obtained the structures with widely tunable mechanical properties. However, an exhaustive search for the optimal one is intractable and computationally expensive. To remedy this, we apply the Artificial Neural Network (ANN) method to achieve the highly efficient multiobjective optimization of the nanocomposites. Since the A_{H} of training samples is tiny (minimum and average values are 0 MJ m^{−3} and 0.51 MJ m^{−3}, respectively) compared with other SMAs, such as TiNi (~10 MJ m^{−3})^{42}, we select the other two mechanical properties (i.e., E_{I} and σ_{L}) for the multiobjective optimization based on the Pareto front^{43}. On the Pareto front, we use a selection strategy to find the three “best” configurations as the candidate and verify them by the phasefield simulation. Our selection strategy transforms the twoobjective optimization problem into a singleobjective optimization problem by calculating the normalized distance δ from a point in search space to the target (E_{I}^{t} = 10 GPa, σ_{L}^{t} = 500 MPa), and the normalized Euclidean distance δ is given by
In order to avoid samples with poor performance dominating the construction of the ML model, we classify the calculated configurations from the HTP phasefield simulations by performing KMeans clustering using the mechanical properties (i.e., E_{I} and σ_{L}) as the input variable. As illustrated in Fig. 4a, the calculated configurations are grouped into three data sets, in which Cluster 1 shows poor mechanical performance in terms of its large E_{I} and low σ_{L}. We thus select Cluster 0 and Cluster 2 (324 samples) as training data for the construction of ANN model. As shown in Supplementary Figure 8, all four features are independent of each other and have certain correlations with the mechanical properties. Moreover, since the chemical features (i.e., F_{Nb} and M_{Nb}) are sampled linearly while the geometric features (i.e. N_{F} and V_{F}) are sampled squarely in the HTP phasefield simulations, we treat the geometric features by square root transformation as \(N_{{{\mathrm{F}}}}^{{{\mathrm{L}}}} = \sqrt {N_{{{\mathrm{F}}}}}\) and \(V_{{{\mathrm{F}}}}^{{{\mathrm{L}}}} = 64 \times \sqrt {V_{{{\mathrm{F}}}}}\) to eliminate the influence of the numerical differences on the prediction performance of the ML model. In order to predict new configurations with better properties, a large unexplored search space is created by increasing the resolution of four structural features: We allow F_{Nb} to vary from 5.0 to 10.0% with an interval of 0.2% and M_{Nb} to vary from 15.0 to 20.0% with an interval of 0.2%. For geometric features, we set \(N_{{{\mathrm{F}}}}^{{{\mathrm{L}}}}\) from 2 to 8 in steps of 2 and \(V_{{{\mathrm{F}}}}^{{{\mathrm{L}}}}\) from 8 to 40 in steps of 4. Overall, 42588 virtual samples (\(C_{26}^1C_{26}^1C_7^1C_9^1\)) are formed out of which only 324 are computationally studied.
Figure 4bc show the tenfold crossvalidation performance of E_{I} and σ_{L} obtained by the ANN algorithm. As we can see, the ANN model has a great crossvalidation performance for E_{I} (R = 0.9889) and σ_{L} (R = 0.9536). According to the hyperparameters determined by the crossvalidation, we use all 324 training data samples to train two new ANN models to predict the E_{I} and σ_{L} of samples in the unexplored search space. According to our selection strategy, three new configurations with improved properties are screened out in the search space, which are selected as promising nanocomposite configurations for the next set of phasefield simulations. Their predicted and calculated mechanical properties are shown in Table 1. The predicted E_{I} is higher than the calculated value with an average prediction error of 1.86%, while the predicted σ_{L} is higher than the calculated value with an average prediction error of 9.46%, indicating that the ML model yield accurate predictions for the unexplored nanocomposites. The calculated properties of the three recommended configurations (denoted as red pentagons) are compared with those from HTP phasefield simulations (denoted as blue dots) in Fig. 4d. Improvements can be observed for the recommended configurations and the red pentagon sample with the most obvious performance improvement is selected as the configuration of the ideal orthopedic implant. Compared with the result of HTP phasefield simulations, the normalized Euclidean distance δ is reduced from 0.231 to 0.200, decreasing by 13.4%. Thus, we could locate the configuration with the best compromise of low E_{I} and large σ_{L} with the aid of ANN. The optimized configuration (as shown in Fig. 5b) is composed of Nbrich TiNb matrix (M_{Nb} = 18.4%) and 25.0% volume fraction of Nblean TiNb square fillers (F_{Nb} = 7.2%) with N_{F} = 16, which will be discussed in detail below. It is also worth to mention that the application of the proposed datadriven approach based on the HTP phasefield simulations is not confined to the design of the present orthopedic implant with low E_{I} and large σ_{L}, but applies to a wide range of materials and structures with target mechanical and other functional properties.
Ideal orthopedic implant design strategy
The comparison of E_{I} and ε_{L} of the constructed nanocomposites designed from the MLaided simulation with those of Ti2448 (Ti24Nb4Zr8Sn in wt.%)^{44}, a typical metallic biomaterial, and other stateoftheart systems proposed in the literature^{45}, are shown in Fig. 5a. It is readily seen that the candidate nanocomposite for the ideal orthopedic implant exhibits an ultralow E_{I} relative to other reported systems, which is essential for metallic biomaterials. The nanocomposite also outperforms Ti2448 in terms of the relatively large elastic strain limit and small hysteresis area. The SS curve of the designed nanocomposite during the loadingunloading cycle together with that of uniform bulk Ti2448 is shown in Fig. 5c with the red curve and gray curve, respectively. The SS curve of Ti2448 features a typical characteristic of a large stress hysteresis with an obvious stress plateau, which is consistent with the experimental observations. By contrast, the stress plateau completely disappears in the SS curve of an ideal orthopedic implant, and exhibits a quasilinear elasticity and almost hysteresisfree mechanical behavior. As shown in Fig. 5a, compared with the stateoftheart functional material^{16}, such as Ti2448 (Ti24Nb4Zr8Sn in wt.%), NICSMA (nanowire in situ composite with SMA), Gum metals et al., the nanocomposite designed by HTP calculations and ML are superior in mechanical properties in terms of their linear superelasticity, ultralow Young’s modulus, large apparent elastic stress limits, and nearhysteresisfree. These properties in the nanocomposites designed by MLassisted HTP calculations are comparable with the recently reported phasefield simulation results via concentration modulation and concentration gradient film designs^{20,21,22,23}.
To gain a thorough understanding of the exceptional mechanical properties, the microstructure evolutions corresponding to the marked points of the loadingunloading SS curve are analyzed, and the results are displayed in Fig. 5d. As shown Fig. 5e, abundant martensitic particles are observed in the nanofillers, while the matrix still presents an austenite phase when the nanocomposite is quenched to 300 K in a stressfree configuration. These observations are because the martensitic phase is more stable in the Nblean fillers (F_{Nb} = 7.2%), as illustrated in Fig. 3a. The emerged temperatureinduced martensitic particles are composed of multivariants in selfaccommodating domain patterns. These martensitic particles behave as seeding beds of martensite, eliminating the nucleation barriers for MT. Upon loading, the stressinduced martensite reorientation occurs, and the favored variants (such that V2, V3, and V4) dominate and gradually fill the fillers. These martensitic variants gradually and continuously grow into the matrix, which is significantly different from the common avalanchelike discontinuous MT. With increasing stress (Fig. 5g and h), the existing martensitic variants continue to expand in the matrix, which is also accompanied by the formation and merging of new martensitic configurations in the matrix. At the end of the loading (600 MPa), the martensitic domains spread almost over the entire area of the nanocomposite with a volume fraction of 94.4%, as illustrated in Fig. 5i, in which the remaining parent phase mainly originates from the martensitic variant interfaces. In the process of unloading, these remaining parent phases initiate the inverse martensite to austenitic transition in the matrix, and the austenite gradually spread from the matrix towards the interfaces of the nanocomposite (Fig. 5j). The martensite completely disappears in the matrix when the applied stress is reduced to 75 MPa (Fig. 5k). In addition, the Nblean fillers also transform back to the initial selfaccommodating multivariants martensite configurations after unloading, which contributes to the zeroresidual strain in a loadingunloading cycle. Therefore, the nanocomposite experiences macroscopically continuous MT throughout the loading and unloading procedure rather than a sharp firstorder transition as that in typical SMAs, which thus exhibits linear superelasticity and ultralow E_{I} (Fig. 5d). The continuous characteristics of the forward and backward MTs originate from the embedded nanofillers with local heterogeneity that induce nonuniform local stress field associated with the geometrical structure and the variation of MT critical stress due to the change of C_{Nb} as discussed above. For a given nanocomposite, nonuniform stress field appears under a certain applied load, as shown in Supplementary Figure 6. The variation in local stresses tailors the total freeenergy landscape with multiple local minima and facilitates the pseudohighorder MT in the nanocomposite.
These results thus suggest a new route for achieving desirable mechanical properties to meet the requirement of different applications by nanocomposite engineering. In particular, the designed TiNb nanocomposite realizes a perfect combination of ultralow modulus, linear superelasticity, and nearhysteresisfree, which cannot be obtained in common materials and will enable many new advanced applications. For instance, the achieved ultralow E_{I} in the current TiNbbased alloys without compromising their good biocompatibility and superior corrosion resistance is of particular importance for biomedical applications. The mismatch in the Young’s modulus between the common implant materials (~110 GPa) and human bone usually causes stress shielding, leading to bone degradation and implant loosening originated from inhomogeneous stress distribution between the implant and the adjacent bone. The elastic modulus of 16.13 GPa in the designed nanocomposite match closely to that of human bones and thus is promising in overcoming this stressshielding issue completely. To verify the functional stability of the designed nanocomposite, the repeated loadingunloading cycles are performed for the ideal orthopedic implant. The second SS curve and the corresponding evolution of microstructures are illustrated in Supplementary Figure 1 in the Supplementary information, which coincide with those in the first one, indicating the mechanical stability of the designed ideal orthopedic implant. In addition, we perform calculations of nanocomposites with organized (triangular and honeycombarray) and random arrays of nanofillers. Supplementary Figure 2 in the Supplementary information shows the simulation results, viz., the SS curves, corresponding microstructures and their evolutions. The SS curves indicate that the mechanical behavior of the nanocomposite is also insensitive to the nanofiller distribution profile. Moreover, we investigate the effect of different shapes of nanofillers on the mechanical properties of the designed nanocomposite. As shown in Supplementary Figure 3 in the Supplementary information, the SS curves of the nanocomposites with square, hexagonal, and spherical nanofillers are almost coincide, indicating that the mechanical properties of the nanocomposite are insensitive to the shape of nanofillers. Last but not least, we also verify that the results remain almost unchanged within the loading rate range of 20 MPa s^{−1} – 500 MPa s^{−1}, as shown in Supplementary Figure 4 in the Supplementary information. Experimental preparation of the proposed fine nanocomposite is a great challenge currently; yet, recent fabrications of size and compositioncontrolled nanocomposites using colloidal synthesis method, replacement reaction, and accumulative roll bonding (ARB) suggest the promising strategies to pursue^{40,46,47}. The combination of HTP phasefield simulations and the ML approach thus provides an efficient paradigm for designing target properties of shape memory alloys, which will also be beneficial to a wide range of functional materials.
Discussion
In conclusion, we have developed a HTP phasefield simulations to accelerate the design of microstructures with desired mechanical properties. HTP phasefield simulations are performed for various TiNb nanocomposites designed by four structural feature variables and it is found that the mechanical responses of the nanocomposites are widely tunable and variable. Based on the combination of HTP phasefield simulations and ML techniques, we design a TiNb alloy nanocomposite with ultralow modulus, linear superelasticity, and nearly free hysteresis, which is promising for biomaterial applications. These superior mechanical properties attribute to the nonuniform stress distribution and the modulation of the critical stress that facilitates continuous MT. The present approach developed by the combination of phasefield simulations and ML is thus an important extension to the previous method of concentration modulations to regulate the MTs, jointly to provide theoretical guidance for the experiments.
Methods
Phasefield model
The TiNbbased alloys undergo diffusionless and reversible MT between the cubic βaustenite (point group \(m\bar 3m\)) and orthorhombic α”martensite (point group mmm) phases^{48} during the thermomechanical loading, as shown in Fig. 1a. This cubictoorthorhombic MT is associated with six variants^{49,50}. Since ML demands a large amount of simulation results, twodimensional phasefield simulations within the plane stress condition are conducted in the present work to illustrate the developed methodology of phasefield simulations integrated with ML. As a result, there are four possible variants (V1–V4) corresponded to four different SFTS tensors after dropping the outofplane strain. All the SFTSs can be found in the Supplementary information. In the phasefield model, a continuous structure order parameter η_{i} (i = 1 − 4) is employed to characterize the MT, with η_{i} = 0 and η_{i} = ±1 representing austenitic and ith martensitic variant, respectively. The total freeenergy of the system F consists of the Landau freeenergy f_{ch}, the gradient energy f_{gr}, and the elastic energy f_{el}: ^{51}
The microstructure evolution of the MT is governed by the timedependent GinzburgLandau (TDGL) equation:^{52}
where L is the kinetic parameter, and ζ_{i} (r,t) is the StochasticLangevin noise term accounting for the effect of thermal fluctuations^{53}. All the phasefield simulation parameters employed in the present work are from Ref. ^{20} and listed in Supplementary Table 2 in the Supplementary information.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
The work is supported by the National Key R&D Program of China (No. 2018YFB0704404) and the National Natural Science Foundation of China (Grant Nos. 11802169 and 12172370).
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Y.Q.Z. and T.X. equally contributed to this work. T.Y.Z. and T.X. supervised and conceived the project. T.X. designed the simulation model. Y.Q.Z. conducted the simulations with J.W.M. and the machine learning with Q.H.W., H.X.Y., H.R.Z., T.S., and T.K. discussed the results. Y.Q.Z., T.X., and T.Y.Z. wrote the manuscript.
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Zhu, Y., Xu, T., Wei, Q. et al. Linearsuperelastic TiNb nanocomposite alloys with ultralow modulus via highthroughput phasefield design and machine learning. npj Comput Mater 7, 205 (2021). https://doi.org/10.1038/s41524021006747
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DOI: https://doi.org/10.1038/s41524021006747
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