Spatially Programmable Architected Materials Inspired by the Metallurgical Phase Engineering

Programmable architected materials with the capabilities of precisely storing predefined mechanical behaviors and adaptive deformation responses upon external stimulations are desirable to help increase the performance and the organic integration of materials with surrounding environments. Here, a new approach inspired by the physical metallurgical principles is proposed to allow the materials designers to not only enhance the global strength but also precisely tune mechanical properties (such as strength, modulus, and plastic deformation) locally in architected materials to create a new class of intelligent mechanical metamaterials. Such programmable materials not only have high strength and plastic deformation stability but also the ability to regulate the local deformation states and spatially control the internal propagation of deformation. This innovative approach also provides new and effective ways to enhance the adaptivity of the materials thanks to responsive strengths that not only make the materials increasingly stronger but also allow threshold‐based adaptive responses to external loading.


Introduction
Architected lattice materials built on a deliberate arrangement of lattice unit cells can achieve unprecedented mechanical properties, such as high strength-to-weight ratio, superior energy absorption capacity, [1] negative Poisson's ratio, [2] negative thermal expansion, [3] programmed morphing routes, [4] and extraordinary damage tolerance, [5,6] which would not be possible via the intrinsic compositions and microstructure of the base materials. [7,8][11][12][13][14][15][16] However, conventional lattice materials DOI: 10.1002/adma.202305846 with high periodicity (single unit cell orientation) often suffer severe stress collapse due to the formation of shear bands [5,6,17,18] during post-yield plastic deformation (i.e., post-yield instability), particularly for stretching-dominated lattices that possess higher strength than bending-dominated lattices at same relative density. [19]][24][25] While the elastic moduli of multi-phase lattice materials can be effectively estimated via the rule of mixture or the Hashin-Shtrikman laws, [26,27] such laws are not capable of describing plastic deformation.Accurate prediction of the plasticity is necessary in programming the post-yield deformation.Therefore, it is important to develop a framework assisting in programming the plastic deformation properties of architected materials.Previous studies demonstrate that the mimicry of crystalline microstructure, known as the meta-crystal approach, enables the translation of key metallurgical strengthening mechanisms to the design of architected materials. [5,6,18,28]Such mesostructures bridge the length scales of the intrinsic microstructure of crystals and mechanical/structure components.In physical metallurgy, the multiphase engineering approach is used to achieve optimal combinations of strength, ductility, and energy absorption of metallic alloys for structural applications in demanding environments. [29][32] Despite of different nature of bondings (physical beams vs atomic bonds), meta-crystals behave similar to natural crystals thanks to the same crystallographic symmetry and topology, and a series of our prior works shows that key principles of physical metallurgy are still applicable for meta-crystals. [5,6,18,28]Therefore, it is possible to use metallurgical constitutive models of crystalline alloys to design metacrystals instead of the rule of the mixture or Hashin-Shtrikman law. [26,27]It should be noted that while the previous studies of the crystal-inspired approach demonstrate novel ways of achieving high strength and post-yield stability, they did not explore the programmability. [5,6,18,28]e here hypothesize that translating the physical metallurgical science (in particular multiphase engineering) to the design of architected materials can provide us with solid design principles with a high degree of confidence to develop materials with highly programmed plastic deformation properties.Guided by metallurgical phase engineering, we encode defined strength and plastic deformation responses to specific locations in architected materials via tailoring features of different phases such as types, spacing, BORs, and volume fractions.Most importantly, due to the high dependency of mechanical properties of meta-crystals on their internal structures [33][34][35][36] (Figure S1, Supporting Information), the encoding of such crystalline phase-mimicking features in meta-crystals could enable a new way of mechanical programmability, in particular the writing of given mechanical responses to specific locations, controlling the load transfer and propagation of deformation inside the material, and programming the plastic flow stress.In distinction to previous reports in programmable mechanical metamaterials made of flexible materials that adversely compromise the strength and post-yield behavior, [37][38][39][40][41][42] herein, we reported crystalline phase-mimicking architected materials that are both structurally strong, spatially programmable, and responsive.

Programmable Mechanical Behavior
To encode a prior knowledge of mechanical properties in crystalinspired architected materials, it is necessary to understand the constitutive responses of multi-phase crystalline alloys, which are governed by the characteristics of their constituent phases. [29]he precipitation strengthening due to the presence of a nondeformable secondary phase (that is different from the matrix) of alloys can be described by a phenomenological model, such as the Orowan hardening constitutive law, [29,43] stating that the strength  p associated with the precipitate phase is proportional to the shear modulus  and inversely to the spacing of the nondeformable phase particles (L − 2r), that is where L is the distance between two second-phase particles, r is the radius of particles, and b is the magnitude of the Burgers vector.
Although the nature of bonding and underlying mechanisms of accommodating the plasticity are different between crystallike architected materials and natural crystals, the mimicry of the crystalline phase uses the same crystallographic symmetry and connectivity, resulting in similar phenomena. [5,6,18,28]That allows the translation of constitutive models describing metallurgical phenomena, such as the Orowan hardening law, to provide a basis for design.Guided by the physical metallurgy of multiphase crystals, we hypothesize that the strengthening of architected materials containing multiphases is governed by the shear modulus of precipitate phase () that is governed by the lattice type, the radius of precipitates (r), distance between precipitates (L), and BOR.Here, we first demonstrated an ability to achieve improved global constitutive response whilst encoding unique mechanical properties locally inside architected materials.Two different meta-crystals containing the same two lattice types: facecentered cubic (FCC) and octet-truss (OT) with the same volume fraction (50% each) and distribution were created with the BOR considered (Figure 1a and Figures S1 and S2, Supporting Information).The rule of mixture and the Hashin-Shtrikman bounding estimates should predict the identical behaviors (e.g., elastic modulus and yield strength) for the two meta-crystals. [26,27]evertheless, the Orowan constitutive law suggests that the behavior of the two meta-crystals should be different depending on the spacing of the lattice phase (either FCC or OT).Indeed, the FCC/OT2 with a reduced spacing had higher strength than FCC/OT1 (Figure 1b,c), suggesting the Orowan law is more applicable (than the rule of mixture or the Hashin-Shtrikman law) in predicting the deformation response of architected materials containing multi-phases.In addition, compared to the single OT phase, that is, a strong phase and exhibits intermittent decreases in flow stress caused by the successive formation of shear bands [44] (Figure S1c, Supporting Information), the two multiphase meta-crystals in Figure 1b show improved plastic behavior with no compromise and much more stable post-yield behavior.It is noteworthy that the FCC/OT2 even outperformed the hard OT single phase regarding the energy absorption capacity (Figure 1c).Such outperformance is significant because it indicates the multi-phase approach can enhance the global mechanical properties of architected materials such as energy absorption and load-bearing capacities.These results confirmed the validity of the use of metallurgical hardening mechanisms to design high-performance architected materials. [5,6,18,28]he use of multiphase engineering to guide the variation of phases not only improves the global deformation behavior but also provides opportunities to program the local response.The mechanical properties, such as Young's modulus, stress distribution, yield strength, and plastic flow stress, of a crystal-like architected phase are characteristically defined by the architecture of its lattice unit cell (Figure S1, Supporting Information).Therefore, deliberately designing and distributing constituent lattice phases enable spatially writing given information of mechanical properties to specific locations, hence encoding (and subsequently programming) the spatial mechanical behaviors inside architected materials.To reveal the effect of encoded information on the plastic deformation behavior, finite element analysis (FEA) and digital image correlation (DIC) (Figure 2) were analyzed.The analyses show that regions filled with softer phases tend to plastically yield first, which is consistent with what was observed in metallurgy: soft phases accommodate the plastic deformation. [45]oth FEA and DIC results demonstrate that the given Young's modulus and yield strength were successfully written and stored in specific locations in meta-crystals to effectively control the local stress/strain states.If a stress threshold was set to be a given value (e.g., 32 MPa, Figure 2e-h), regions experiencing stress above (or below) the threshold can be seen as ON (or OFF).Alternatively, it can be considered as 1 (or 0) state-analogous to the writing of binary information in computers.Figure 2 shows that the change in lattice spacing effectively controls the local states.In particular, the FCC/OT2 with the finest spacing between phases exhibited not only the highest and more stable mechanical performance (Figure 1b) but also highly regulated stress (Figure 2f vs e) and strain local states (Figure 2j vs i).Additional controllability can be achieved by varying not only the spacing but also the lattice type.We therefore replaced the OT in the FCC/OT1 with the BCC to create FCC/BCC1 having the same volume, size, and distribution as those of FCC/OT1 (Figure 1a and Figure S2, Supporting Information).Figures 1b and 2a,c show that the change of the phase type significantly altered the global and local states of plastic deformation behavior (e.g., FCC/OT1 vs FCC/BCC1).
Such highly accurate controllability of the local deformation state inside architected materials under the external loading (Figure 2a-h) provides an ability to program the deformation and failure paths.Upon loading, the plastic deformation was dictated to be first concentrated at the softer phase thanks to its lower load-bearing capacity (i.e., FCC in FCC/OT and BCC in FCC/BCC, Figure 2i-l), then transferred to the hard phase once the soft phase collapsed, similar to the strain partition in dualphase steels. [46]The local plastic deformation accommodated by struts buckling was formed along the shear system of lattice: shear planes of {022} in BCC [47] and {002} in FCC. [6]The soft phase spatially distributed throughout an architected structure allowed the plastic deformation (and failure) to only propagate through to the adjacent domains of the same soft phase.In particular, the four designs of meta-crystals (Figure 1a) created four different means to propagate the plastic deformation path.In detail, the plastic deformation only propagated in the FCC domains at the beginning of the deformation of the two FCC/OT metacrystals (Figure 2i,j) in consistent with the simulation prediction (Figure 2a,b).In contrast to severely localized strain in FCC/OT1, the FCC/OT2 exhibited a more homogeneous and highly regulated pattern of plastically deformed and non-plastically formed (i.e., ON and OFF) states (Figure 2f vs e), similar to the ON/OFF stress states shown in Figure 2e-h.Although such ON/OFF plastic deformation states were not reversible in this study, it is possible to make them reversible by combining the bistable structural mechanisms.Moreover, such finer and homogeneous plastic deformation avoided severe strain localization which is often a main source of damage.The controlling of propagation of plastic deformation bands thanks to the deliberate arrangement of hard/soft phases is responsible for the steady stress response and enhanced energy absorption of lattices seen in Figure 1b,c.Compared to FCC/OT1, FCC/BCC1 showed that the replacement of OT by BCC changed the main load bearer to be FCC and deformation accommodation to the BCC phase, thereby propagation via the BCC phase in FCC/BCC1 instead of FCC (Figure 2k vs i).Shear bands accommodated by BCC domains propagated step-wise instead of mostly horizontally like in FCC/OT1.In addition to the well-defined elastic stress patterns under small deformation (Figure 2a-h), these different phase combinations also showed that the strain localization in plastic deformation regime can be regulated in a defined manner and sequence via phase engineering thanks to the defined mechanical properties of individual phases (Figure S1, Supporting Information).Because of the properties' dependency on the structures of lattice, [20,21] the alternation of strain localization from FCC in FCC/OT to BCC in FCC/BCC highlights the effectiveness of utilizing means of phase engineering to program not only elastic but also plastic deformation with a high confidence, which contrasts with the uncertainty associated with processing defects that would compromise the programmability of architected materials. [48,49]he deformation path in FCC/BCC2 was more singular along a Ω-like path highlighting the effect of distribution and morphology of the secondary phase that was nonuniformly distributed (Figure 2l) on the propagation of mechanical deformation.After the collapse of BCC domains, the plastic deformation was bridged via multiple small shear bands in the FCC phases.It is worth noting that the load redistribution caused by the collapse of the softer phase activated different shear band activities, [6,22] resulting in different deformation of the FCC phase in the two FCC/BCC meta-crystals (Figure 2k,l).Moreover, before the full collapse of the global lattice structure, the soft phase can act as a cushion shielding the hard phase from external load.This implies that the soft phases can be programmed to protect important objects (such as embedded sensors) inside the materials.The defined stress, or strain, during deformation, can also be used for diagnosis and monitoring the safety status of architected components when bearing load. [5]Therefore, the phase engineering approach provides an effective means to not only enhance mechanical strength but also program deformation behaviors and dictate the way in which the mechanical behavior propagates between different domains of phases inside materials.It is worth noting that metallurgical phase engineering includes tailoring the volume fraction, phase types, the BOR, morphology, and distribution. [50]Because the metallurgical principles are demonstrated to be highly applicable to engineer architected materials as demonstrated earlier, in Section S3, Supporting Information, and in previous studies, [5,6,18,28] it provids multiple ways of guiding the encoding and programming of plastic deformation behaviors of architected materials both globally and locally.

Programming the Constitutive Deformation Responses
A conventional single-phase architected material usually exhibits a single mode of mechanical response (e.g., Figure S1, Supporting Information, where elastic behavior is followed by an apparently plateau plastic regime with a single level of strength before the densification).In engineering applications, it is often desirable that a material can have multiple levels of strength to precisely respond to varying external loads.A stepwise behavior is also necessary to write and perform Boolean operations on the basis of threshold (i.e., threshold function).Consequently, such a stepwise behavior significantly enhances the programmability and functionality of materials.Here we showed that it is possible to design multiphase meta-crystals that have a precisely defined strength  as an increasingly smooth step function of constituent phase (p), strength () of each constituent phase and strain (ɛ), that is where p refers to architected phases with specific internal structures.p i = (p 1 , p 2 , …, p n ) with n is the number of constituent phases in the meta-crystal,  is phase p determined strength. i = ( 0 ,  1 , …,  n ) with  0 = 0 and  i (with i > 0) is the strength of the phase p i in the post-yield plateau region, and is the global compressive strain.ɛ i = (ɛ 0 , ɛ 1 , …, ɛ n ) with ɛ 0 = 0 and ɛ i (i > 0) is a midpoint of strain in a smooth transition i between the two strengths  i − 1 and  i .(ɛ i − e i ) and (ɛ i + e i ) are the beginning and the end of the transition i (i.e., e i is the half of the strain range over which the transition occurs and e 0 = 0).In other words,  is a smooth and increasing step function with a step height of  i over a strain range of [ɛ i + e i , ɛ i + 1 − e i + 1 ], which is determined by the arrangement of different phases.It is observed in metallurgy that the lamellar microstructure of multiphases is effective in controlling the deformation behavior thanks to the strengths of constituent phases.One example is TiAl alloys which are widely used in engineering applications. [51]TiAl can contain a lamellar microstructure of face-centered tetragonal (FCT) -TiAl and hexagonal-close packed (HCP)  2 -Ti 3 Al (Figure 3a,b).The lamellar microstructure governs the localization (due to slip and twinning) and internal stress reaction at the interfaces between the two phases, hence the directionality of response under external loading. [52]Mimicking a multiphase crystalline lamellar microstructure at a mesoscale can not only allow the designer to guide the internal transfer of mechanical load but also enable stepwise multi-strength levels and smooth transition between strengths, that is, a smooth staircase strength response.
We designed a dual-phase meta-crystal inspired by lamellar / 2 microstructure including key metallurgical features of the two phases, such as the BOR, that is, {111}  ∥ {0001}  2 , found in TiAl alloys (Figure 3a,b).The meta-crystal was comprised of two phases (i.e., n = 2): phase 1 (p 1 ) -a hexagonal-close-packed structure HCP mimicking  2 , and phase 2 (p 2 ) -a FCC (mimicking the  -it is worth noting that the FCT crystal structure of -TiAl is very similar to an FCC crystal). [53]The two phases were in the lamellar form and arranged alternatively (Figure 3c and Figure S6, Supporting Information).The used HCP is weaker than FCC (Figure 3d) because partly of its lower relative density  r_HCP = 0.218 in comparison to that of FCC (  r_FCC = 0.395).The strength  1 of the HCP was about 3 MPa, and  2 of the FCC was about 11 MPa.FEA revealed that the ⟨ 1 2 0 1 2 1 2 ⟩ struts of HCP phases were the main load bearer while that of the FCC was the 〈011〉 struts (Figure 3e), which were in consistent with those of their single phases (Figure S12, Supporting Information) thanks to the coherent nodal connection across their interfaces.The HCP phase, as expected, yielded first due to its lower yield strength than that of the FCC phase (Figure 3f), dictating the first strength level of the meta-crystal, that is,  1 =  1 = 3 MPa over a strain range of [4%, 33%] giving ɛ 1 ≈ 2%, e 1 ≈ 2%, ɛ 2 ≈ 37%, and e 2 ≈ 4% (Figure 3d).The collapse of the HCP layers (over a strain range of 2e 2 = 8%) led to the transition from  1 = 3 MPa to  2 = 11 MPa.The strain range over which a strength level is active is defined by the layer thickness of the corresponding phase (Figures S9 and S10, Supporting Information).The evolution of stress response with strain exhibited by the HCP/FCC metacrystal exhibits high reproducibility of such multi-step stress responses (Figure S13, Supporting Information) and follows the written mechanical properties of constituent phases.Although there are some negligible fluctuations of strength in each level of stress, such fluctuations are considerably smaller compared to the distinct difference between the two strength levels.Therefore, the fluctuations do not affect the accuracy of Boolean operations based on the stress threshold.
As discussed before, another way to program the multistrength is to vary the density of internal structures (i.e., functionally graded density structures) of single-phase meta-crystals (Figure S7, Supporting Information).However, it is more difficult to control the propagation of mechanical load along specific directions due to the uniformity of the lattice orientation.In addition, a crystalline phase is not only defined by the crystal structure but also by the atomic bonding.For example, the gamma phase in Nickel superalloys is chemically disordered FCC of Ni and alloying elements.Replacing this chemically-disordered FCC arrangement of atoms with a chemically ordered FCC between Ni and Al (hence the atomic bonding) leads to the L1 2 phase (i.e., gamma') that provides significant strength at high temperatures. [54,55]Because the atomic bonds are mimicked by the physical struts in architected materials, varying the strut diameter or the node spacing (as done in the functionally graded density approach) results in a different phase.Therefore, the functionally graded lattice can be considered as part of the crystalline multiphase-inspired approach.By varying both the strut diameter and lattice types, another combination of BCT and HCP phases inspired by dualphase titanium alloys (e.g., Ti6Al4V alloy) [56] was also designed (Figure 3g and Figure S8, Supporting Information), extending the level of stepped strength to three active over a range of deformation (Figure 3h).

Conclusion
We developed an innovative approach inspired by the multiphase engineering in metallurgy to develop architected lattice materials that are not only strong and tough but also have predefined mechanical behavior and programmed different local states inside materials activated upon loading.The approach also enables programmed multi-strength behavior (namely, a smooth step-like function) to allow the materials to perform adaptive bearing ability and threshold-based logical operation.We presented a solid foundation of design on the basis of the multiphase hardening mechanisms (such as the Orowan mechanism and deformability of secondary particles) to strengthen meta-crystals and control the local behaviors via deliberately distributing different architected phases to specific locations.We demonstrated the clear similarities between crystalline alloys and architected lattice materials and the additional basis for understanding the behaviors of multi-phase meta-crystals via metallurgy mechanisms, both in the yield strength and hardening behavior in post-yield plastic deformation.The deliberate phase engineering in metacrystals guided by metallurgy mechanisms also enables regulating local states (such as plastically deformed/undeformed equivalent to ON/OFF) throughout the materials under loading.To enable more genuine programmability and expand their application as mechanical components, this study also creates architected materials that behave in a smooth step functional manner to convert the external loading to multi-level stress responses.Such materials were inspired by lamellar microstructure found in titanium alloys (such as TiAl and Ti6Al4V).The number of response levels can be achieved by incorporating the functionally graded structures into the crystal-inspired approach.It is worth noting that the crystal-inspired approach not only enables the ability to write tailored mechanical properties to specific locations but also results in hardening, improved stability (after the yielding), and enhanced energy absorption, significantly improving the global mechanical performance, achieving both functional and structural properties.
On the basis of the phase engineering in physical metallurgy, we also demonstrated that it is very straightforward and accurate in tailoring the writing of information and the programming of response via engineering the phase characteristics such as the lattice structure type, lattice spacing, phase dimensions/distribution, and phase volume fractions.The engineering of phase is underpinned by the wealth of knowledge achieved in metallurgy (such as the Orowan equation), helping to achieve high confidence in engineering and designs and allowing more programming flexibilities of mechanical information across different length scales to develop genuinely smart and programmable architected materials for a variety of applications requiring both high strength and stability and logical thinking functionalities.

Figure 1 .
Figure 1.Dual-phase architected materials and their mechanical responses.a) Dual-phase meta-crystals: FCC/OT1, FCC/OT2, FCC/BCC1, and FCC/BCC2.b) Corresponding stress-strain curves of the four meta-crystals.c) Yield stress and energy absorption up to 60% strain of dual-phase meta-crystals and their constituent phases: FCC, OT, and BCC (note: both yield stress and energy absorption were normalized by their relative densities).

Figure 2 .
Figure 2.Activation and propagation of mechanical information of dual-phase meta-crystals (i.e., FCC/OT1, FCC/OT2, FCC/BCC1, and FCC/BCC2) upon compression loading.a-d) Stress distributions at 1.5% strain from FEA. e-h) Stress patterns with a stress threshold applied at 1.5% strain from FEA. i-l) Experimental strain distributions at 15% global compression strain from DIC analysis.

Figure 3 .
Figure 3. Smooth step function response.a,b) Crystal structures of -TiAl and  2 -Ti 3 Al with the highlighted crystal planes corresponding to the / 2 interface.c) Lamellar / 2 -inspired meta-crystal containing HCP and FCC phases.d) Stress-strain curves of single HCP and FCC phases, and the HCP/FCC meta-crystal that shows two-stepped strength.The error bar of HCP/FCC is the standard deviation calculated from three repetitive tests (Figure S13, Supporting Information).e) Stress distribution at 3% nominal strain of the HCP/FCC from FEA. f) Local deformation field revealed by digital image correlation analysis (the color bar represents the maximum shear strain).g) Meta-crystal containing two HCP phases (of different strut diameters) written alternatively with a BCT phase, the top insets showing the BOR.h) Black solid curve: Three-stepped strength responses; dashed curves: the stress-strain curves of the three single phases.