The Emergence of Sequential Buckling in Reconfigurable

materials

Article The Emergence of Sequential Buckling in Reconﬁgurable Hexagonal Networks Embedded into Soft Matrix

Pavel I. Galich 1 , Aliya Sharipova 2 and Slava Slesarenko 3,*

1 Department of Aerospace Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel; [email protected] 2 Institute of Strength Physics and Materials Science, SB RAS, 634055 Tomsk, Russia; [email protected] 3 Faculty of Mathematics and Mechanics, Saint Petersburg State University, 198504 Saint Petersburg, Russia * Correspondence: [email protected]

Abstract: The extreme and unconventional properties of mechanical metamaterials originate in their sophisticated internal architectures. Traditionally, the architecture of mechanical metamaterials is decided on in the design stage and cannot be altered after fabrication. However, the phenomenon of elastic instability, usually accompanied by a reconfiguration in periodic lattices, can be harnessed to alter their mechanical properties. Here, we study the behavior of mechanical metamaterials consisting of hexagonal networks embedded into a soft matrix. Using finite element analysis, we reveal that under specific conditions, such metamaterials can undergo sequential buckling at two different strain levels. While the first reconfiguration keeps the periodicity of the metamaterial intact, the secondary buckling is accompanied by the change in the global periodicity and formation of a

new periodic unit cell. We reveal that the critical strains for the ﬁrst and the second buckling depend on the metamaterial geometry and the ratio between elastic moduli. Moreover, we demonstrate that

Citation: Galich, P.I.; Sharipova, A.; the buckling behavior can be further controlled by the placement of the rigid circular inclusions in Slesarenko, S. The Emergence of the rotation centers of order 6. The observed sequential buckling in bulk metamaterials can provide Sequential Buckling in additional routes to program their mechanical behavior and control the propagation of elastic waves. Reconﬁgurable Hexagonal Networks Embedded into Soft Matrix. Materials Keywords: mechanical metamaterials; buckling; sequential buckling; instabilities; elastic wave 2021, 14, 2038. https://doi.org/ propagation; reconﬁguration 10.3390/ma14082038

Academic Editor: Antonio Santagata 1. Introduction Received: 17 March 2021 Accepted: 13 April 2021 Cellular lattice structures assembled from repeating cells are widely known for their Published: 18 April 2021 superior combination of low weight and high mechanical properties [1–3]. In general, the mechanical performance of lattices is deﬁned by the geometry of the unit cells [4–7], and

Publisher’s Note: MDPI stays neutral rational design of the internal architecture enables the engineering of lattices with enhanced with regard to jurisdictional claims in stiffness [8,9], controlled anisotropy of mechanical properties [10,11], or lattices that are published maps and institutional afﬁl- tolerant to partial failure [12,13]. Unsurprisingly, many mechanical metamaterials can be iations. considered as successors of cellular materials [14,15]. Indeed, the unconventional properties of mechanical [16–18], elastic [19–23], and acoustic metamaterials [24,25] originate in their intrinsic periodicity, along with the rational design of unit cells. Simultaneously, such intrinsic periodicity can give rise to the elastic instability phenomena often observed in cellular materials and mechanical metamaterials [26]. While Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. for engineering materials loss of stability is usually unwanted, on par with failure or This article is an open access article delamination, for functional metamaterials, loss of stability can be frequently harnessed distributed under the terms and to control their unconventional properties [26–31]. The elastic instabilities were used to conditions of the Creative Commons adjust the stiffness or auxeticity of structured materials [25,32], open and close elastic Attribution (CC BY) license (https:// bandgaps [33], and realize other unusual wave phenomena [34]. Adding a soft deformable creativecommons.org/licenses/by/ matrix into the design facilitates extra coupling between stiff components, enabling more 4.0/).

Materials 2021, 14, 2038. https://doi.org/10.3390/ma14082038 https://www.mdpi.com/journal/materials Materials 2021, 14, 2038 2 of 13

involved buckling behavior accompanied by the formation of various instability-driven patterns [35,36]. The classical and perhaps the simplest examples of such systems in 2D and 3D are periodic layered and fiber composites, respectively. Under uniaxial compression, such composites can undergo elastic loss of stability at a specific value of critical strain defined mainly by the composite geometry and the elastic modulus contrast [37–40]. In layered and fiber composites, instabilities may develop at different wavelengths ranging from the size of a typical heterogeneity to the size of an entire specimen [41,42]. The onset of macroscopic (or long-wave) instabilities, characterized by a wavelength significantly larger than a characteristic microstructure size, can be predicted by the loss of ellipticity analysis with the homogenized tensor of elastic moduli [43–45] that in certain cases can be obtained via micromechanics-based homogenization [46–48]. Analysis of microscopic instabilities with wavelengths comparable to the period of a structure requires the application of more involving techniques, such as Bloch–Floquet analysis [42,49]. Surprisingly, the Bloch– Floquet method helped to demonstrate that the secondary loss of stability may occur in bilayer structures under some specific conditions [50]. However, to the best of our knowledge, a realization of such sequential buckling was reported only in the stiff films attached to the soft substrates [51–53] or in multilayered structures embedded into the soft media [54,55]. Note that, here, we do not consider cases when the sequential loss of stability occurs due to the change in the metamaterial topology because of contact between elements [56]. Recently, Gao et al. [57] showed that metamaterials consisting of stiff hexagonal networks embedded into a soft matrix undergo loss of stability under biaxial compression. Similar to layered or fiber composites, the instability-induced patterns can be classified as Type 1 or Type 2, corresponding to microscopic and macroscopic instabilities, respectively. In the post-buckling regime, the amplitude of instability-induced patterns increases with an increase in the applied strain, while the overall structure remains the same. It has been shown that such instability-induced transformations can be harnessed to open elastic bandgaps with strain-controlled characteristics. At the same time, the maximal level of applied strain for equibiaxial compression does not exceed 10% for Type 1 and 16% for Type 2 modes. One can notice that in the post-buckling regime, the dispersion curves for metamaterials with Type 1 buckling [57] have values close to zero for the specific wavevectors at the contour of the irreducible Brillouin zone (IBZ). In layered composites, such behavior foreshadows the oncoming loss of stability [34]. An instability occurs when a zero eigenvalue is found for a non-trivial wavevector. In other words, instabilities are associated with zero modes other than rigid body translations and rotations. Zero modes have been observed in Kagome lattices [58] and origami structures [59] and have even been harnessed for elastic cloaking [60]. In the current manuscript, we will explore the buckling phenomenon and demonstrate that stiff hexagonal networks can undergo secondary loss of stability for some specific geometries and material constants. Moreover, we will demonstrate how critical strains for the primary and secondary buckling can be controlled using stiff inclusions placed in the rotation centers of order 6.

2. Materials and Methods Figure1a shows a periodic hexagonal network embedded into a soft deformable matrix. The strut thickness t and the width of a hexagonal element H univocally deﬁne the geometry of a metamaterial, while the additional parameter ri is necessary for a metamaterial with central inclusions (Figure1b). The struts and matrix were considered to be hyperelastic materials with the neo-Hookean strain energy density function

2 W = 0.5µ(I1 − 3) − µ ln(J) + 0.5λ(ln(J)) ,

where I1 and J are the ﬁrst and third invariants of the right Cauchy–Green deformation tensor, and µ and λ are Lame constants that can be expressed through Young’s modulus Materials 2021, 14, 2038 3 of 13

Materials 2021, 14, x FOR PEER REVIEW 3 of 15

E Eν where 𝐼 and 𝐽 are the= first and third invariants= of the right Cauchy–Green deformation E and Poisson’s ratio ν as µ 2(1+ν) and λ (1+ν)(1−2ν) , respectively. Subscripts N and tensor, and 𝜇 and 𝜆 are Lame constants that can be expressed through Young’s modulus M stand for network𝐸and Poisson’s and ratio matrix, 𝜈 as 𝜇= respectively. and 𝜆= The Poisson, respectively. ratios Subscripts of both N and materials were () ()() assumed to beM equal,stand for namely,network andν Mmatrix,= ν respectively.N = 0.3, whileThe PoissonEN ratios/EM of> both1. materials The materials were had the same densitiesassumedρN = toρ beM .equal, namely, 𝜈 =𝜈 =0.3, while 𝐸/𝐸 >1. The materials had the same densities 𝜌 =𝜌.

Figure 1. Hexagonal networks embedded into a soft deformable matrix without (a) and with (b) central inclusions. A Figure 1. Hexagonal networks embedded𝒓 𝒓 into a soft deformable matrix without (a) and with (b) primitive unit cell is shown in red color. Lattice vectors and couple the points on the opposite boundaries of the Commented [M2]: Is the bold necessary? primitive unit cell. central inclusions. A primitive unit cell is shown in red color. Lattice vectors rH and rV couple the Commented [PG3R2]: Yes, but only for r points on the opposite2.1. Search boundariesfor Sequential Instabilities of the primitive unit cell. To facilitate a search for the multiple onsets of instability and analyze the propaga- 2.1. Search for Sequential Instabilities tion of elastic waves, we used finite element (FE) analysis in COMSOL 5.4. The employ- Commented [M4]: Pleas define if appropriate. To facilitatement a of search the FE method for the for multiplecomputation onsetsof dispersion of instability relations in periodic and analyze materials relies the propagation on the approach developed in [61,62]. The following multistep procedure, similar to [50], Commented [PG5R4]: of elastic waves,was used we to used study finite the reconfiguration element (FE)of metamaterials analysis undergoing in COMSOL sequential 5.4. instabili- The employment of the FE methodties. for computation of dispersion relations in periodic materials relies on the approachA. developed Search for the in first [61 onset,62]. of The instabilities following multistep procedure, similar to [50], was 1. We identified the primitive unit cell in the undeformed state (Figure 1a). used to study2. the We reconfiguration subjected the selected of unit metamaterials cell to equibiaxial compression undergoing under sequential plane strain con- instabilities. A. Search forditions the first by applying onset ofthe instabilitiesfollowing periodic boundary conditions: 𝑢 =𝑢 −𝜀𝐻 𝐻 ⎧ 1. We identified the primitive unit𝑣 cell=𝑣 in the undeformed state (Figure1a). ⎪ ⎪ 𝜋 2. We subjected the selected𝑢 unit=𝑢 cell−𝜀 to𝐻 equibiaxial 𝐻 cos compression under plane 3 strain conditions by applying⎨ the following periodic𝜋 boundary conditions: ⎪𝑣 =𝑣 −𝜀𝐻 𝐻 sin ⎪ 3 ⎩ 𝑢 =𝑣 =0  h Commented [M6]: 𝐻 may need be changed to where 𝑢 and 𝑣are horizontal anduright vertical= u displacements,le f t − εH respectively, 𝐻 𝐻 and 𝐻  𝐻  are horizontal and vertical periods (𝐻vright=𝐻 ==𝐻v le for f tan initial state), and 𝜀 is the applied strain (positive for compression). = − v π 3. For the obtained deformedutop state, weu performedbottom a εsweepH cos along3 the perimeter of the IBZ com-  𝜔(𝒌) 𝜔 v π puting dispersion relationsvtop =, vwherebottom −is theεH eigenfrequencysin 3 for the corresponding wavevector 𝒌 at the IBZ contour [63]. To this end, Bloch–Floquet conditions were superim- uA = vA = 0 posed on the finitely deformed metamaterials using the following equations on the primitive Commented [PG7R6]: yes! unit cell boundaries: where u and v are horizontal and vertical displacements, respectively, Hh and H v are h v horizontal and vertical periods (H = H = H for an initial state), and ε is the appliedCommented [M8]: Is the bold necessary?

strain (positive for compression). Commented [PG9R8]: yes 3. For the obtained deformed state, we performed a sweep along the perimeter of the IBZ computing dispersion relations ω(k), where ω is the eigenfrequency for the corresponding wavevector k at the IBZ contour [63]. To this end, Bloch–Floquet conditions were superimposed on the ﬁnitely deformed metamaterials using the following equations on the primitive unit cell boundaries:

 u = u e−ikrH  right le f t  −ikrH  vright = vle f te u = u e−ikrV  top bottom  −ikrV vtop = vbottome

where rH and rV are the vectors connecting matching points on horizontal and vertical boundaries of the cell. These equations were implemented in COMSOL using integrated Bloch–Floquet conditions. 4. If ω(k) > 0 for all wavevectors k except for the trivial one k = (0, 0), then the material remains stable at the speciﬁed strain level ε. Materials 2021, 14, 2038 4 of 13

5. We repeated steps 2–4, gradually increasing the applied strain ε by 0.01% until the 1 1 1 1 non-trivial kcr with ω kcr = 0 was found. The determined εcr = ε and kcr are the critical strain and critical eigenmode, respectively, corresponding to the first or primary onset of instabilities. 6. For verification of the buckling mode, we also performed linear buckling analysis [57] to compare critical strains and buckling modes found using these two methods. B. Subsequent buckling 7. We used the previously found buckling modes as an initial imperfection to guide a reconfiguration of the metamaterial after reaching the buckling strain. If buckling was accompanied by the periodicity change, we increased the unit cell using Floquet continuation. The amplitude of the superimposed imperfection was chosen to be 1/1000 of the stiff layer thickness. The imperfections were superimposed in COMSOL with the help of the Deformed Geometry (dg) interface. 8. We continuously increased the applied strain ε beyond the onset of the first buckling strain and observed the instability-driven transformation of the structure with a gradually increasing buckling amplitude. 9. When necessary, we defined a new primitive cell and updated the IBZ to perform procedure A again, searching for the second onset of instability. Theoretically, this procedure can be repeated indefinitely to search for sequential onsets of instabilities.

2.2. Wave Propagation Analysis The propagation of the plane elastic waves through the metamaterial was studied using a similar approach due to intricate relations between elastic waves and elastic instabilities [42,63]. Bloch–Floquet boundary conditions (step A3) were superimposed on the ﬁnitely deformed metamaterial, and the corresponding eigenproblem was solved for wavevectors k at the IBZ contour [57].

3. Results 3.1. Hexagonal Networks Embedded into Soft Matrix First, we recall the classification of two types of buckling modes that can be observed in hexagonal networks embedded into the soft deformable matrix [57]. Similar to the micro-instability and macro-instability, we can distinguish Type 1 buckling with local wrinkling patterns (Figure2c) and Type 2 buckling accompanied by the formation of a global alternating pattern (Figure2b). For clarity, we define Type 1 buckling as a case when the metamaterial periodicity remains unaltered, while Type 2 is defined as a case when a new primitive unit cell must be constructed. Figure2a shows the map of the eigenmodes as a function of the thickness to the cell ratio α = t/H and the elastic modulus contrast β = EN/EM. As it was already established for a large enough α and β, Type 2 buckling is realized, while for relatively small values, the metamaterial buckles with the formation of a wrinkling pattern. In general, Type 2 1 buckling occurs when parameter γ = αβ 3 > γcr, defined as

! 1 3 − 4ν 3 γcr = 2.17 (1) (1 − ν)2

Note that this analytical estimation (obtained previously in [57]) is shown here for the speciﬁc case of linear elastic constituents with equal Poisson’s ratios νM = νN = ν. The continuous solid line in Figure2a corresponds to Equation (1), and it should separate (α, β) pairs associated with Type 1 and Type 2 buckling. However, multiple simulations for various (α, β) pairs reveal that this expression underestimates the threshold value of γcr, and microscopic buckling is observed even for γ & γcr (see the blue stars above the Materials 2021, 14, 2038 5 of 13

black curve). This slight discrepancy is partially caused by the fact that in Equation (1), the struts of hexagonal networks are treated as Euler–Bernoulli beams. We observed that with an increase in the elastic modulus contrast, the critical value of γ observed in the simulations tends to the critical value defined by Equation (1); however, this is beyond the scope of the current manuscript. While for γ γcr metamaterials undergo buckling with the formation of fine wrinkling patterns, here, we focus on the case γ ≈ γcr associated with Type 1 buckling in which a pattern that accommodates half of the critical wavelength can be observed (Figure2d). (α, β) combinations corresponding to the formation of this specific pattern are shown by the blue star symbol in Figure2a. Such a pattern still satisfies the definition of Type 1 buckling since there are no changes in the periodicity. However, the vicinity of such metamaterials to the configurations with global buckling (Type 2) implies that critical buckling strains associated with Type 1 and Type 2 buckling may be very close

Materials 2021,to 14, x each FOR PEER other, REVIEW enabling conditions for the subsequent loss of stability. Therefore,5 of 15 we will further focus mainly on the buckling behavior of these intermediate Type 1.5 metamaterials.

Figure 2. (a) Buckling modes as a function of geometrical 𝛼 and material 𝛽 parameters. (b) Type 2 global mode denoted by solidFigure black squares. 2. (a ()c)Buckling Type 1 local mode modes denoted as aby function empty red circles. of geometrical (d) Type 1.5 bucklingα and mode material consideredβ parameters. in this (b) Type 2 study denotedglobal by mode blue stars. denoted The solid by continuous solid black line corresp squares.onds (toc )the Type analytical 1 local estima modetion of denoted the transition by zone empty red circles. (d) between Type 1 and Type 2 buckling modes, i.e., Equation (1). Type 1.5 buckling mode considered in this study denoted by blue stars. The solid continuous line corresponds to theFigure analytical 2a shows estimation the map of ofthe theeigenmodes transition as a zonefunction between of the thickness Type 1to andthe cell Type 2 buckling 𝛼=𝑡/𝐻 𝛽=𝐸 /𝐸 modes, i.e., Equationratio (1). and the elastic modulus contrast . As it was already established for a large enough 𝛼 and 𝛽, Type 2 buckling is realized, while for relatively small values, the metamaterial buckles with the formation of a wrinkling pattern. In general, Type 2 Figure3c,dbuckling illustrate occurs when the behaviorparameter 𝛾 of:=𝛼𝛽 elastic >𝛾 waves, definedin as the mechanical metamaterialCommented with [M17]: Please check if it is correct α = 1/30 and β = 1000 in the vicinity of the first onset of elastic instability. In particular, 3−4𝜈 Commented [PG18R17]: it's correct (1) Figure3c shows the evolution of the lowest𝛾 =2.17 branches of dispersion curves observed as (1−𝜈) the applied strainNoteε increases. that this analytical One estimation can see (obtained that an previously increase in in [57]) the is appliedshown here strain for leads to a decrease in thethe phasespecific case velocity of linear of elastic elastic constituents waves. with In equal physics, Poisson’s this ratios phenomenon 𝜈 =𝜈 =𝜈. The is also known as mode “softening”continuous solid [64 ].line According in Figure 2a corresponds to the procedure to Equation (1), described and it should above, separate metamaterial (𝛼, 𝛽) pairs associated with Type 1 and Type 2 buckling. However, multiple simulations remains stablefor whenvarious (ω𝛼,(k) 𝛽 ) pairs> 0revealfor that all thisk. Noteexpression that underestimates for k = (0, the 0 threshold), there value is always of a trivial solution ω(k)𝛾=, and0, associatedmicroscopic buckling with theis observed rigid even body for movement. 𝛾 ≳𝛾 (see the At blue the stars same above time, the sinceCommented Type [M19]: Please check if it is correct black curve). This slight discrepancy is partially caused by the fact that in Equation (1), 1 buckling does not cause the change in periodicity, to find the onset of Type 1 buckling,if it should we be “≥”？ the struts of hexagonal networks are treated as Euler–Bernoulli beams. We observed that track the second lowest shear branch of the dispersion diagram, focusing on a non-trivial with an increase in the elastic modulus contrast, the critical value of 𝛾 observed in the Commented [PG20R19]: it's correct simulations tends to the critical value defined by Equation (1); however,loc this is beyond solution for k = (0, 0). Figure3d shows how the eigenvalue ω = ω(G) computed for vector k = (0,the 0 )scope, corresponding of the current manuscript. to point WhileG of forthe 𝛾≪𝛾 IBZ metamaterials contour, decreases undergo buckling with an increase with the formation of fine wrinkling patterns, here, we focus on the case 𝛾≈𝛾 associ- loc 1, loc in the appliedated strain with Type until 1 buckling it reaches in whichω a crpattern= 0thatfor accommodates the strain halfεcr of the critical= 0.87% wave-. This critical strain correspondslength can to be the observed first onset (Figure of 2d). the (𝛼, instabilities 𝛽) combinations associated corresponding with to the Type formation 1 local buckling, of this specific pattern are shown by the blue star symbol in Figure 2a. Such a pattern still and the correspondingsatisfies the definition buckling of Type mode 1 buckling is shown since inthere Figure are no 4changesa. Surprisingly, in the periodicity. simultaneously, the eigenfrequencyHowever,ω thegl vicinity= ω( ofK )suchcomputed metamaterials for to point the configurationsK of the with IBZ global contour buckling also noticeably (Type 2) implies that critical buckling strains associated with Type 1 and Type 2 buckling 1, loc Commented [M21]: Is the italics necessary? decreases withmay the be very increase close to each in ε ,other, especially enabling conditions in the vicinity for the subsequent of εcr loss. The of stability. pattern shown in Figure4b correspondsTherefore, we towill point further Kfocus, and mainly this on patternthe buckling has behavior a different of these periodicityintermediate comparedCommented [PG22R21]: yes Type 1.5 metamaterials. gl 1, loc to the undeformed state. However, ω > 0 for εcr ; therefore, Type 1 buckling occurs

Materials 2021, 14, x FOR PEER REVIEW 6 of 15

Figure 3c,d illustrate the behavior of elastic waves in the mechanical metamaterial with 𝛼=1/30 and 𝛽 = 1000 in the vicinity of the first onset of elastic instability. In particular, Figure 3c shows the evolution of the lowest branches of dispersion curves observed as the applied strain 𝜀 increases. One can see that an increase in the applied strain leads to a decrease in the phase velocity of elastic waves. In physics, this phenomenon is also known as mode “softening” [64]. According to the procedure described above, metamaterial remains stable when 𝜔(𝒌) >0 for all 𝒌. Note that for 𝒌=(0,0), there is always a trivial solution 𝜔(𝒌) =0, associated with the rigid body movement. At the same time, since Type 1 buckling does not cause the change in periodicity, to find the onset of Type 1 buckling, we track the second lowest shear branch of the dispersion diagram, focusing on a non-trivial solution for 𝒌=(0,0). Figure 3d shows how the eigenvalue 𝜔 𝜔 = Commented [M23]: May 𝜔 be changed 𝜔(𝑮) computed for vector 𝒌=(0,0), corresponding to point G of the IBZ contour, de- to 𝜔 ? Materials 2021, 14, 2038 creases with an increase in the applied strain until it reaches 𝜔 =0 for the strain 6 of 13 , 𝜀 =0.87%. This critical strain corresponds to the first onset of the instabilities associated with Type 1 local buckling, and the corresponding buckling mode is shown in Figure 4a. Surprisingly, simultaneously, the eigenfrequency 𝜔 =𝜔(𝑲) computed for point K of the IBZ contour also noticeably decreases with the increase in 𝜀, especially in the vicin- , ﬁrst. Realizationity of of𝜀 buckling. The pattern should shown in lead Figure to 4b thecorresponds change to point in the K, and geometry this pattern of has the unit cell; , however, if noa different perturbations periodicity arecompared superimposed to the undeformed on thestate. undeformed However, 𝜔 >0 state, for 𝜀 the; numericalCommented [PG24R23]: yes! therefore, Type 1 buckling occurs first. Realization of buckling should lead to the change

methods mayin fail the to geometry detect of the the bucklingunit cell; however, mode, if no allowing perturbations us toare reachsuperimposed higher on strains the without any reconﬁgurations.undeformed state, the numerical methods may fail to detect the buckling mode, allowing Commented [25]: us to reach higher strains without any reconfigurations.

Figure 3. The primitive unit cell (a) and its reciprocal lattice (b) of the studied structures in the undeformed state. The gray areaFigure represents 3. Thethe irreducible primitive Brillouin unit zone cell (IBZ (a), )and and the red its arrows reciprocal show the lattice path along (b its)of contour. the studied(c) Evolu- structures in the tion ofundeformed lowest branches of state. dispersion The curves gray in the area vicinity represents of the primary the instability. IBZ, and * denotes the redthe minimal arrows strain show (𝜀= the path along its 0.87%) for which a non-trivial zero eigenvalue is found at point G. (d) Evolution of the eigenfrequencies corresponding to Typecontour. 1 and Type (c 2) modes Evolution with an ofincrease lowest in the branches applied strain. of dispersion The Y-axis represents curves normalized in the vicinity frequency of 𝑓= the primary instability. *. denotes the minimal strain (ε = 0.87%) for which a non-trivial zero eigenvalue is found at point G. (d) Evolution of the eigenfrequencies corresponding to Type 1 and Type 2 modes with an increase in Materials 2021, 14, x FOR PEER REVIEW q 7 of 15 the applied strain. The Y-axis represents normalized frequency f = ωH ρM . 2π µM

Figure 4. Type 1 (a) and Type 2 (b) patterns associated with the first onset of buckling. Figure 4. Type 1 (a) and Type 2 (b) patterns associated with the first onset of buckling. Figure 5a shows the dispersion diagrams for metamaterials in undeformed (𝜀=0%) Figure5aand shows deformed the ( dispersion𝜀=7%) states. diagramsNote that the fordispersion metamaterials curves for the deformed in undeformed state are ( ε = 0%) obtained for two distinct cases—with (Figure 5c) and without (Figure 5b) reconfiguration, and deformed (ε = 7%) states. Note that the dispersion, curves for the deformed state are triggered by the first onset of instabilities at 𝜀 =0.87%. One can see that without re- obtained for twoconfiguration, distinct the cases—with dispersion curves (Figure continuously5c) and move without towards (Figurethe x-axis,5 enablingb) reconfiguration, us , to define 𝜀 =0.90% as the lowest strain level,1, loc such as 𝜔 =0 (see Figure 3d). The triggered by the first onset of instabilities at εcr = 0.87%, . One can see that without linear perturbation analysis confirms that this critical value 𝜀 and the corresponding reconfiguration,eigenmode the dispersion (Figure 4b) can curves be observed continuously as the second minimal move buckling towards mode the if thex-axis, en- enabling 1, gl us to define ε larged= 0.90%unit cellRVEas, the consisting lowest of nine strain primitive level, unit such cells, is as considered.ωgl = 0 However,(see Figure the 3d).Commented The [26]: cr 𝜀, second value can be reached only because the instability-driven1, gl transformation as- linear perturbationsociated analysiswith Type 1 confirmsbuckling was that restricted. this Figure critical 5b reveals value thatεcr if theand reconfiguration the corresponding Commented [U27]: Here K capital does not occur, after the onset of local instability in the post-buckling regime, 𝜔(𝒌) =0 eigenmode (Figure4b) can be observed as the second minimal buckling mode ifFormatted: the English (United States) for the selected directions 𝒌, and, therefore, the phase velocity of the waves propagating enlarged unit cell,through consisting the metamaterial of nine is equal primitive to zero unit[65]. Zero cells, phase is considered. velocity corresponds However, to the the second 1, gl degenerate case, and the existence of zero eigenvalues after the bifurcation hints at incon- value εcr can be reached only because the instability-driven transformation associated sistencies in numerical analysis. In contrast, if reconfiguration takes place, zero eigenfre- with Type 1 bucklingquencies are was not observed restricted. apart from Figure trivial5 onesb reveals (Figure 5c), that and the if themetamaterial reconfiguration main- does not occur, aftertains the its onset stability of in localthe updated instability configuration. in the post-buckling regime, ω(k) = 0 for the selected directions k, and, therefore, the phase velocity of the waves propagating through the metamaterial is equal to zero [65]. Zero phase velocity corresponds to the degenerate case, and the existence of zero eigenvalues after the bifurcation hints at inconsistencies in numerical analysis. In contrast, if reconfiguration takes place, zero eigenfrequencies are not observed apart from trivial ones (Figure5c), and the metamaterial maintains its stability in the updated configuration.

Figure 5. Dispersion curves for metamaterial with 𝛼=1/30 and 𝛽 = 1000. (a) Undeformed state, (b) 𝜀 = 7% without reconfiguration, (c) 𝜀 = 7% with reconfiguration.

Figure 6 shows the pattern evolution after the first buckling. With an increase in the applied strain 𝜀, the amplitude of the pattern gradually increases (compare Figure 6a,b), while the node positions continue to form the right hexagons, proving that the metamaterial keeps its initial periodicity even in the post-buckling regime. However, according to Figure 7c, with a further increase in the applied strain, the value 𝜔 starts to tend to zero

Materials 2021, 14, x FOR PEER REVIEW 7 of 15

Figure 4. Type 1 (a) and Type 2 (b) patterns associated with the first onset of buckling.

Figure 5a shows the dispersion diagrams for metamaterials in undeformed (𝜀=0%) and deformed (𝜀=7%) states. Note that the dispersion curves for the deformed state are obtained for two distinct cases—with (Figure 5c) and without (Figure 5b) reconfiguration, , triggered by the first onset of instabilities at 𝜀 =0.87%. One can see that without reconfiguration, the dispersion curves continuously move towards the x-axis, enabling us , to define 𝜀 =0.90% as the lowest strain level, such as 𝜔 =0 (see Figure 3d). The , linear perturbation analysis confirms that this critical value 𝜀 and the corresponding eigenmode (Figure 4b) can be observed as the second minimal buckling mode if the enlarged unit cellRVE, consisting of nine primitive unit cells, is considered. However, the , Commented [26]: second value 𝜀 can be reached only because the instability-driven transformation associated with Type 1 buckling was restricted. Figure 5b reveals that if the reconfiguration Commented [U27]: Here K capital does not occur, after the onset of local instability in the post-buckling regime, 𝜔(𝒌) =0 Formatted: English (United States) for the selected directions 𝒌, and, therefore, the phase velocity of the waves propagating through the metamaterial is equal to zero [65]. Zero phase velocity corresponds to the degenerate case, and the existence of zero eigenvalues after the bifurcation hints at incon- Materials 2021, 14, 2038 7 of 13 sistencies in numerical analysis. In contrast, if reconfiguration takes place, zero eigenfrequencies are not observed apart from trivial ones (Figure 5c), and the metamaterial maintains its stability in the updated configuration.

Figure 5.Figure Dispersion 5. Dispersion curves for metamaterial curves for with metamaterial 𝛼=1/30 and with𝛽 = 1000α =. (a1) /Undeformed30 and β =state,1000 (b) .(𝜀a) = 7% Undeformed without state, reconfiguration, (c) 𝜀 = 7% with reconfiguration. (b) ε = 7% without reconfiguration, (c) ε = 7% with reconfiguration. Figure 6 shows the pattern evolution after the first buckling. With an increase in the Figure6 showsapplied thestrain pattern 𝜀, the amplitude evolution of the after pattern the gradually first buckling. increases With(compare an Figure increase 6a,b), in the applied strainwhileε, the the amplitude node positions of continue the pattern to form graduallythe right hexagons, increases proving (compare that the metamate-Figure6a,b ), rial keeps its initial periodicity even in the post-buckling regime. However, according to while the nodeFigure positions 7c, with continuea further increase to form in the the applied right strain, hexagons, the value proving 𝜔 starts thatto tend the to zero metamaterial keeps its initial periodicity even in the post-buckling regime. However, according to Figure7c, with a further increase in the applied strain, the value ωG starts to tend to 2,gl zero again. Eventually, ωG = 0 for the value of strain ε = 10.8% (Figure7c ). Since Materials 2021, 14, x FOR PEER REVIEW cr 8 of 15 the zero eigenfrequency is observed for point K, the metamaterial undergoes secondary buckling accompanied by global reconfiguration with the formation of the same pattern as , 1, loc for hypotheticalagain. global Eventually, buckling 𝜔 =0 canceled for the value by the of strain first local𝜀 = reconfiguration 10.8% (Figure 7c). atSinceεcr the zero= 0.87% (Figure4b ). However,eigenfrequency since is observed the local for patternspoint K, the have metamaterial already undergoes developed secondary prior buckling to the sec- Commented [28]: accompanied by global reconfiguration with the formation of the same pattern as for hy- ond onset of the instabilities, the resulting patterns in the post-buckling, regime (after pothetical global buckling canceled by the first local reconfiguration at 𝜀 = 0.87% Commented [29]: or secondary buckling)(Figure 4b). gain However, a mixed since geometry the local patterns combining have already the developed features prior from to boththe second buckling modes (Figureonset8 ). of This the instabilities, sequential the buckling resulting pattern lowerss in the the post-buckling plane group regime of the (after metamaterial second- from p6mm inary the buckling) undeformed gain a mixed state geometry down to combinin p3 (e.g.,g the see features [66] for from a referenceboth buckling regarding modes the (Figure 8). This sequential buckling lowers the plane group of the metamaterial from plane groups).p6mm With in the the furtherundeformed increase state down in the to appliedp3 (e.g., see strain [66] forε, thea reference formed regarding pattern the contin- ues to evolve.plane At the groups). same With time, the onefurther can increase see that in the the applied applied strain strain 𝜀, the doesformed not pattern significantly con- affect the dispersiontinues to curvesevolve. At after the same the time, first one reconfiguration can see that the applied (Figure strain6d,e), does and not significantly the secondary buckling leadsaffect to the the noticeabledispersion curves increase after the in first the reconfiguration density of dispersion (Figure 6d,e), curves and the (Figure secondary6f) due buckling leads to the noticeable increase in the density of dispersion curves (Figure 6f) to the more complexdue to the geometry more complex of thegeometry new of primitive the new primitive cell. cell.

Figure 6. Evolution of the pattern (a-c) and corresponding dispersion curves (d-f) after the first and the second buckling Commented [30]: for metamaterialFigure 6.withEvolution 𝛼 = 1/30 ofand the 𝛽 pattern = 1000. (a (, ad–) c𝜀=3%) and, corresponding (b, e) 𝜀=9%, (c, dispersionf) 𝜀 = 25% (updated curves IBZ (d– isf) shown after thein ﬁrst and Figure 7b).the Colors second denote buckling von Mises for stresses metamaterial (color ranges with are notα =synchronized1/30 and forβ (=a–c1000)), (d) .(###;a, d(e)) ###;ε = (f3%) ###..,(b ,e) ε = 9%,(c,f) Commented [31]: ? ε = 25% (updated IBZ is shown in Figure7b). Colors denote von Mises stresses (color ranges are not Formatted: Font: Bold, Complex Script Font: Bold synchronized for (a–c)). Formatted: Font: Bold, Complex Script Font: Bold Formatted: Font: Bold, Complex Script Font: Bold Formatted: Font: Bold, Complex Script Font: Bold

Figure 7. The primitive unit cell (a) and its reciprocal lattice (b) of the studied structure after secondary buckling. The updated (virtual) IBZ contour is shown in red [66]. (c) Evolution of the lowest eigenfrequency at point K in the vicinity of primary instability.

Materials 2021, 14, x FOR PEER REVIEW 8 of 15

, again. Eventually, 𝜔 =0 for the value of strain 𝜀 = 10.8% (Figure 7c). Since the zero eigenfrequency is observed for point K, the metamaterial undergoes secondary buckling Commented [28]: accompanied by global reconfiguration with the formation of the same pattern as for hy- , pothetical global buckling canceled by the first local reconfiguration at 𝜀 = 0.87% Commented [29]: or (Figure 4b). However, since the local patterns have already developed prior to the second onset of the instabilities, the resulting patterns in the post-buckling regime (after secondary buckling) gain a mixed geometry combining the features from both buckling modes (Figure 8). This sequential buckling lowers the plane group of the metamaterial from p6mm in the undeformed state down to p3 (e.g., see [66] for a reference regarding the plane groups). With the further increase in the applied strain 𝜀, the formed pattern con- tinues to evolve. At the same time, one can see that the applied strain does not significantly affect the dispersion curves after the first reconfiguration (Figure 6d,e), and the secondary buckling leads to the noticeable increase in the density of dispersion curves (Figure 6f) due to the more complex geometry of the new primitive cell.

Figure 6. Evolution of the pattern (a-c) and corresponding dispersion curves (d-f) after the first and the second buckling Materials 2021, 14, 2038 8 of 13 Commented [30]: for metamaterial with 𝛼 = 1/30 and 𝛽 = 1000. (a, d) 𝜀=3%, (b, e) 𝜀=9%, (c, f) 𝜀 = 25% (updated IBZ is shown in Figure 7b). Colors denote von Mises stresses (color ranges are not synchronized for (a–c)), (d) ###; (e) ###; (f) ###.. Commented [31]: ?

Formatted: Font: Bold, Complex Script Font: Bold Formatted: Font: Bold, Complex Script Font: Bold Formatted: Font: Bold, Complex Script Font: Bold Formatted: Font: Bold, Complex Script Font: Bold

Figure 7. The primitive unitFigure cell ( 7.a) Theand primitiveits reciprocal unit lattice cell (a )(b and) of its the reciprocal studied latticestructure (b) after of the secondary studied structure buckling. after The updated (virtual) IBZ contoursecondary is shown buckling. in red [66]. The updated(c) Evolution (virtual) of the IBZ lowest contour eigenfrequency is shown in red at[ 66point]. (c K) Evolution in the vicinity of the of Materials 2021, 14, x FOR PEER REVIEW 9 of 15 primary instability. lowest eigenfrequency at point K in the vicinity of primary instability.

Figure 8. Evolution of the pattern after the second buckling at 𝜀 = 30% (a), 𝜀 = 50% (b). Colors denote von Mises stresses (colorFigure ranges are 8. notEvolution synchronized of for the (a, b pattern)). after the second buckling at ε = 30% (a), ε = 50% (b). Colors denote von Mises stresses (color ranges are not synchronized for (a, b)). We note that all considered Type 1.5 metamaterials (Figure 2a) undergo the subsequent loss of stability. Therefore, one can conclude that for selected 𝛼 and 𝛽 such as We note that all considered Type 1.5 metamaterials (Figure2a) undergo the subsequent 𝛼𝛽 ≈𝛾 , Type 1 local buckling does not cancel the Type 2 instability but barely post- 1 loss of stability.pones Therefore, it. For instance, one for can the concludemetamaterial thatwith for𝛼 = selected 0.05 (𝐻/𝑡α = 20)and and β𝛽such = 350, as αβ 3 ≈ , , γ , Type 1 localglobal buckling buckling occurs does only not after cancel an additional the Type12% strain 2 instability(𝜀 −𝜀 ≈ but 11.7% barely). Figure postpones cr 9 shows the values of the critical buckling strains that correspond to the first and second- it. For instance,ary foronsets the of instability metamaterial for the selected with metamaterialsα = 0.05 denoted(H/t by= smaller20) starand symbolsβ = in350 , global 2, gl, 1,,loc buckling occursFigure only 2a. Due after to the an definition additional of Type 12% 1.5 metamaterials strain (ε 𝜀 −<𝜀ε , one≈ may11.7% notice). Figure9 that the geometric factor 𝛼 does not affect the primary bucklingcr in crany significant way shows the values(Figure of the9b). In critical contrast, buckling an increase strainsin parameter that 𝛼, correspondfor instance, due to to thethe wider first struts and secondary onsets of instabilityof the hexagonal for the network selected postpones metamaterials the secondary denotedbuckling, making by smaller the metamaterial star symbols in more stable. A more significant effect is observed concerning the material1, loc factor1,gl 𝛽. Figure Figure2a . Due9a to reveals the the definition opposite dynamics of Type for the 1.5 onsets metamaterials of the first and secondεcr instabilities.< εcr , oneWhile may notice that the geometricmetamaterials factor withα does a higher not elastic affect modulus the contrast primary undergo buckling primary buckling in any at significantlower way strains, the subsequent secondary buckling is postponed as compared with metamaterials (Figure9b ). In contrast,with a lower anelastic increase modulus in contrast. parameter Hence, theα, strain for instance, range between due two to consecutive the wider struts of the hexagonal networkbucklings is postponeswider for the metamaterials the secondary with a higher buckling, contrast making between the the elastic metamaterial moduli more stable. A moreof significant constituents. For effect example, is observed for the mechanical concerning metamaterial the with material 𝛼=1/30 factor and 𝛽=β. Figure9a 800, the width of such a strain range is 9.7% versus 10.8% for a metamaterial with 𝛽= reveals the opposite1800. In general, dynamics since Type for 1 the buckling onsets induces of fewer the first perturbations and second in the architecture instabilities. of While metamaterialsmechanical with a higher metamaterials, elastic in modulussome practical contrast cases, it may undergo be practical primary to alter the buckling geomet- at lower strains, the subsequentric parameters secondary of the hexagonal buckling networks is with postponed the intention as to broaden compared the operation with metamaterialscon- ditions by postponing the global buckling. However, the observed effect is relatively mi- with a lower elasticnute and modulus requires a precise contrast. selection Hence, of 𝛼 and the 𝛽; therefore, strain rangeeven slight between fabrication two imper- consecutive bucklings is widerfections for may the undermine metamaterials the desiredwith effect. a higher contrast between the elastic moduli of constituents. For example, for the mechanical metamaterial with α = 1/30 and β = 800, the width of such a strain range is 9.7% versus 10.8% for a metamaterial with β = 1800. In general, since Type 1 buckling induces fewer perturbations in the architecture of mechanical metamaterials, in some practical cases, it may be practical to alter the geometric parameters of the hexagonal networks with the intention to broaden the operation conditions by postponing the global buckling. However, the observed effect is relatively minute and requires a precise selection of α and β; therefore, even slight fabrication imperfections may undermine the desired effect.

Materials 2021, 14, 2038 9 of 13 Materials 2021, 14, x FOR PEER REVIEW 10 of 15

Figure 9. Dependencies of theFigure first and 9. Dependenciesthe second buckling of the strains first on and material the second parameter buckling 𝛽 (a) strainsand geometrical on material parame- parameter β (a) and ter 𝛼 (b). geometrical parameter α (b). 3.2. Hexagonal Networks with Embedded Inclusions 3.2. Hexagonal Networks with Embedded Inclusions As the soft matrix in the considered metamaterials couples the deformation of the stiff strutsAs the on the soft unit matrix cell boundaries, in the considered the additional metamaterials modification couplesof the soft the matrix deformation seems of the tostiff be a struts promising on the way unit to tune cell their boundaries, stability and the control additional the propagation modification of elastic of waves. the soft matrix Thus,seems we to modify be a promising the architecture way of to the tune metamaterials their stability by placing and control a stiff circular the propagation inclusion of elastic inwaves. the rotation Thus, center we modify of order the 6—the architecture center of the of soft the hexagon metamaterials (Figure 1b). by placingIt was previ- a stiff circular ouslyinclusion shown, in however, the rotation that center various of systems order 6—the consisting center of rigid of the inclusions soft hexagon embedded (Figure into1 b). It was apreviously soft matrix could shown, undergo however, instability-driven that various transformations systems consisting with the of formation rigid inclusions of wavy embedded patternsinto a soft or twinning matrix could domains undergo for a singular instability-driven loss of stability transformations [67]. Here, we harness with the inclu- formation of sions (without reducing the symmetry group of the metamaterial) as an additional factor, wavy patterns or twinning domains for a singular loss of stability [67]. Here, we harness enabling us to control the values of critical buckling strains. We select the elastic constants ofinclusions the inclusions (without to match reducing the elastic the constant symmetrys of the group stiff network, of the metamaterial) causing the inclusions as an additional tofactor, be virtually enabling undeformable. us to control the values of critical buckling strains. We select the elastic , , , constantsFigure of10 theshows inclusions dependencies to match of the the values elastic of critical constants strains of ( the𝜀 stiff, 𝜀 network,, and 𝜀 causing) the oninclusions the radius to of be the virtually inclusions undeformable. 𝑟 and dimensionless parameter 𝛿=2𝑟/𝐻 for metamate- 𝛼=1/30 𝛽 = 1000 1, loc 1,gl 2,gl rials withFigure 10 shows and dependencies. One can of the see valuesthat relatively of critical small strains inclusions (εcr do, ε notcr , and εcr ) significantly affect the first onset of instabilities, and the metamaterial still undergoes two on the radius of the inclusions ri and dimensionless parameter δ = 2ri/H for metama- subsequentterials with transformations.α = 1/30 and However,β = 1000 with. an One increase can see in the that inclusion relatively radius, small the inclusionsdif- do ference between the two first buckling modes (𝜀, −𝜀,) starts to decrease until 𝜀, > not significantly affect the first onset of instabilities, and the metamaterial still under- 𝜀, approximately when 𝛿≈0.4, representing the transition from the configuration goes two subsequent transformations. However, with an increase in the inclusion radius, with sequential buckling to the configuration with single-step buckling. Simultaneously, 1,gl 1, loc thethe critical difference strain between corresponding the two to the first secondary buckling buckling modes monotonically (εcr − εcr )decreases starts to with decrease until 1,loc 1,gl anεcr increase> εcr inapproximately the inclusion radius. when Finally,δ ≈ 0.4 for, representing 𝛿≈0.4, which the corresponds transition fromto the the transi- configuration , , , tionwith between sequential modes, buckling we have to 𝜀 the configuration=𝜀 =𝜀 . Hence, withsingle-step the addition buckling. of the central Simultaneously, in- clusionsthe critical proves strain to be corresponding an efficient strategy to the to secondary control buckling buckling behavior monotonically in metamaterials decreases with consistingan increase of hexagonal in the inclusion networks radius. embedded Finally, into for a softδ ≈ deformable0.4, which matrix. corresponds to the transition 1, loc 1,gl 2,gl between modes, we have εcr = εcr = εcr . Hence, the addition of the central inclusions proves to be an efficient strategy to control buckling behavior in metamaterials consisting of hexagonal networks embedded into a soft deformable matrix. While embedded inclusions are shown to affect the buckling behavior, they can also alter the propagation of the elastic waves in metamaterials, especially if the density contrast between the soft matrix and inclusions can be realized. Figure 11 shows dispersion curves for the selected metamaterial with α = 1/30, β = 1000, δ = 0.1 deformed up to 10%. One can see that while the dispersion curves for the metamaterials without (Figure 11a) and with (Figure 11b,c) inclusions are very similar, the additional density contrast between embedded inclusions may facilitate the opening of the elastic bandgaps (Figure 11c). Since

the localized masses can be employed to design locally resonant elastic metamaterials, the change in the architecture caused by the elastic instabilities in the hexagonal networks could be potentially harnessed to control bandgaps. Materials 2021, 14, x FOR PEER REVIEW 11 of 15

Materials 2021, 14, 2038 10 of 13 Materials 2021, 14, x FOR PEER REVIEW 11 of 15

Figure 10. Values of critical strains for the hexagonal networks with embedded circular inclusions.

While embedded inclusions are shown to affect the buckling behavior, they can also alter the propagation of the elastic waves in metamaterials, especially if the density contrast between the soft matrix and inclusions can be realized. Figure 11 shows dispersion curves for the selected metamaterial with 𝛼 = 1/30, 𝛽 = 1000, 𝛿 =0.1deformed up to 10%. One can see that while the dispersion curves for the metamaterials without (Figure 11a) and with (Figure 11b,c) inclusions are very similar, the additional density contrast between embedded inclusions may facilitate the opening of the elastic bandgaps (Figure 11c). Since the localized masses can be employed to design locally resonant elastic metamaterials, the change in the architecture caused by the elastic instabilities in the hexago- Figure 10. Values of critical strains for the hexagonal networks with embedded circular inclusions. Figure 10. Valuesnal networks of critical could strains be potentially for the hexagonal harnessed to networks control bandgaps. with embedded circular inclusions. While embedded inclusions are shown to affect the buckling behavior, they can also alter the propagation of the elastic waves in metamaterials, especially if the density contrast between the soft matrix and inclusions can be realized. Figure 11 shows dispersion curves for the selected metamaterial with 𝛼 = 1/30, 𝛽 = 1000, 𝛿 =0.1deformed up to 10%. One can see that while the dispersion curves for the metamaterials without (Figure 11a) and with (Figure 11b,c) inclusions are very similar, the additional density contrast between embedded inclusions may facilitate the opening of the elastic bandgaps (Figure 11c). Since the localized masses can be employed to design locally resonant elastic metamaterials, the change in the architecture caused by the elastic instabilities in the hexagonal networks could be potentially harnessed to control bandgaps.

𝛼 = 1/30, 𝛽 = 1000, 𝛿 =0.1 Figure 11.Figure Dispersion 11. curvesDispersion for the deformed curves forcomposite the deformed with composite with α after= 1 primary/30, β buckling= 1000, andδ = 0.1 after 𝜀 = 10% for metamaterials with no inclusions (a), 𝜌/𝜌 =1 (b), 𝜌/𝜌 =8 (c). The red rectangles denote complete bandgapsprimary for in-plane buckling waves. and ε = 10% for metamaterials with no inclusions (a), ρI /ρM = 1 (b), ρI /ρM = 8 (c). The red rectangles denote complete bandgaps for in-plane waves. 4. Conclusions 4. Conclusions We revisited the mechanical behavior of the stiff hexagonal networks embedded into the soft matrix, primarily focusing on the phenomenon of elastic loss of stability. Previ- We revisitedously, thetwo mechanicaldifferent types behaviorof instability-driven of the stiff transformations hexagonal were networks distinguished embedded de- into the soft matrix,pending primarily on the focusingperiodicity onof the the emerged phenomenon patterns. ofIn elasticthis paper, loss we of revealed stability. that Previously, me- two differentchanical types metamaterials of instability-driven consisting of transformationsstiff hexagonal networks were embedded distinguished into the soft depending ma- on the periodicity of the emerged patterns. In this paper, we revealed that mechanical metamaterials consisting of stiff hexagonal networks embedded into the soft matrix could Figure 11. Dispersion curvesundergo for the deformed sequential composite buckling with 𝛼 for = the1/30, specific 𝛽 = 1000, combinations 𝛿=0.1 after primary of buckling geometric and and material pa- 𝜀 = 10% for metamaterials rameters. with no inclusions Under ( equibiaxiala), 𝜌/𝜌 =1 compression,(b), 𝜌/𝜌 =8 (c these). The red metamaterials rectangles denote first complete undergo local buckling, bandgaps for in-plane waves. keeping their periodicity in the post-buckling regime. However, with the further increase in4. Conclusions the applied strain, a secondary loss of stability accompanied by the formation of new patternsWe revisited can be the triggered. mechanical We behavior studied of the how stiff geometricalhexagonal networks and materialembedded factors into affect the bucklingthe soft matrix, behavior. primarily In particular, focusing on we thedemonstrated phenomenon ofthat elastic the loss elastic of stability. modulus Previ- contrast affects theously, values two different of the criticaltypes of straininstability-driven for primary transformations and secondary were losses distinguished of stability de- in opposite ways:pending while on the theperiodicity metamaterial of the emerged becomes patterns. less stableIn this withpaper, an we increase revealed that in contrast me- between elasticchanical moduli, metamaterials the critical consisting strain of stiff corresponding hexagonal networks to the embedded secondary into buckling the soft ma- increases. The method to search for the sequential instabilities using COMSOL was described. Moreover, we highlighted the importance of the minute perturbations in the undeformed states for numerical analysis to avoid degenerate dispersion curves with zero phase veloci- ties. This analysis helped us to show that even though initially two eigenvalues associated with local and global buckling might be close to each other, the primary loss of stability can significantly postpone the secondary buckling, accompanied by the global reconfiguration. We demonstrated that the placement of additional stiff inclusions in the rotation centers of order 6 can be employed to control the buckling behavior and propagation of elastic waves. In conclusion, we note that the experimental verification of the described phenomena can Materials 2021, 14, 2038 11 of 13

be a challenging task due to the relatively narrow region in the design space for which sequential buckling is observed, and imperfections associated with the manufacturing process [68,69]. Nevertheless, we believe that these results are useful for engineering other bulk systems with sequential buckling.

Author Contributions: Conceptualization, S.S.; methodology, S.S. and P.I.G.; formal analysis, S.S.; investigation, S.S., P.I.G., and A.S.; writing—original draft preparation, S.S and P.I.G.; writing— review and editing, S.S., P.I.G., and A.S.; supervision, S.S.; funding acquisition, S.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Russian Scientific Foundation, grant number 20-71-00017. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References 1. Fleck, N.A.; Deshpande, V.S.; Ashby, M.F. Micro-Architectured Materials: Past, Present and Future. Proc. R. Soc. Math. Phys. Eng. Sci. 2010, 466, 2495–2516. [CrossRef] 2. Schaedler, T.A.; Carter, W.B. Architected Cellular Materials. Annu. Rev. Mater. Res. 2016, 46, 187–210. [CrossRef] 3. Yan, C.; Hao, L.; Hussein, A.; Raymont, D. Evaluations of Cellular Lattice Structures Manufactured Using Selective Laser Melting. Int. J. Mach. Tools Manuf. 2012, 62, 32–38. [CrossRef] 4. Wadley, H.N.G. Multifunctional Periodic Cellular Metals. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 2006, 364, 31–68. [CrossRef] 5. Ajdari, A.; Jahromi, B.H.; Papadopoulos, J.; Nayeb-Hashemi, H.; Vaziri, A. Hierarchical Honeycombs with Tailorable Properties. Int. J. Solids Struct. 2012, 49, 1413–1419. [CrossRef] 6. Berinskii, I.E. Elastic in–Plane Properties of Cellular Materials: Discrete Approach. Mech. Mater. 2020, 148, 103501. [CrossRef] 7. Berinskii, I.E. In-Plane Elastic Properties of Auxetic Multilattices. Smart Mater. Struct. 2018, 27, 075012. [CrossRef] 8. Zorzetto, L.; Ruffoni, D. Re-Entrant Inclusions in Cellular Solids: From Defects to Reinforcements. Compos. Struct. 2017, 176, 195–204. [CrossRef] 9. Chen, Z.; Wang, Z.; Zhou, S.; Shao, J.; Wu, X. Novel Negative Poisson’s Ratio Lattice Structures with Enhanced Stiffness and Energy Absorption Capacity. Materials 2018, 11, 1095. [CrossRef][PubMed] 10. Xu, S.; Shen, J.; Zhou, S.; Huang, X.; Xie, Y.M. Design of Lattice Structures with Controlled Anisotropy. Mater. Des. 2016, 93, 443–447. [CrossRef] 11. Kulagin, R.; Beygelzimer, Y.; Estrin, Y.; Schumilin, A.; Gumbsch, P. Architectured Lattice Materials with Tunable Anisotropy: Design and Analysis of the Material Property Space with the Aid of Machine Learning. Adv. Eng. Mater. 2020, 22, 2001069. [CrossRef] 12. Cherkaev, A.; Ryvkin, M. Damage Propagation in 2d Beam Lattices: 2. Design of an Isotropic Fault-Tolerant Lattice. Arch. Appl. Mech. 2019, 89, 503–519. [CrossRef] 13. Ryvkin, M.; Slesarenko, V.; Cherkaev, A.; Rudykh, S. Fault-Tolerant Elastic–Plastic Lattice Material. Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 2020, 378.[CrossRef][PubMed] 14. Zheng, X.; Lee, H.; Weisgraber, T.H.; Shusteff, M.; DeOtte, J.; Duoss, E.B.; Kuntz, J.D.; Biener, M.M.; Ge, Q.; Jackson, J.A.; et al. Ultralight, Ultrastiff Mechanical Metamaterials. Science 2014, 344, 1373–1377. [CrossRef] 15. Meza, L.R.; Zelhofer, A.J.; Clarke, N.; Mateos, A.J.; Kochmann, D.M.; Greer, J.R. Resilient 3D Hierarchical Architected Metamateri- als. Proc. Natl. Acad. Sci. USA 2015, 112, 11502–11507. [CrossRef] 16. Reinbold, J.; Frenzel, T.; Münchinger, A.; Wegener, M. The Rise of (Chiral) 3D Mechanical Metamaterials. Materials 2019, 12, 3527. [CrossRef] 17. Barchiesi, E.; Spagnuolo, M.; Placidi, L. Mechanical Metamaterials: A State of the Art. Math. Mech. Solids 2019, 24, 212–234. [CrossRef] 18. Scerrato, D.; Giorgio, I. Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deforma- tions. Symmetry 2019, 11, 1523. [CrossRef] 19. Krushynska, A.O.; Miniaci, M.; Bosia, F.; Pugno, N.M. Coupling Local Resonance with Bragg Band Gaps in Single-Phase Mechanical Metamaterials. Extreme Mech. Lett. 2017, 12, 30–36. [CrossRef] 20. Miniaci, M.; Krushynska, A.; Movchan, A.B.; Bosia, F.; Pugno, N.M. Spider Web-Inspired Acoustic Metamaterials. Appl. Phys. Lett. 2016, 109, 071905. [CrossRef] 21. Miniaci, M.; Krushynska, A.; Gliozzi, A.S.; Kherraz, N.; Bosia, F.; Pugno, N.M. Design and Fabrication of Bioinspired Hierarchical Dissipative Elastic Metamaterials. Phys. Rev. Appl. 2018, 10, 024012. [CrossRef] Materials 2021, 14, 2038 12 of 13

22. Chang, S.-Y.; Chen, C.-D.; Yeh, J.-Y.; Chen, L.-W. Elastic Wave Propagation of Two-Dimensional Metamaterials Composed of Auxetic Star-Shaped Honeycomb Structures. Crystals 2019, 9, 121. [CrossRef] 23. Krushynska, A.O.; Galich, P.; Bosia, F.; Pugno, N.M.; Rudykh, S. Hybrid Metamaterials Combining Pentamode Lattices and Phononic Plates. Appl. Phys. Lett. 2018, 113, 201901. [CrossRef] 24. Ma, G.; Sheng, P. Acoustic Metamaterials: From Local Resonances to Broad Horizons. Sci. Adv. 2016, 2, e1501595. [CrossRef] 25. Liu, J.; Guo, H.; Wang, T. A Review of Acoustic Metamaterials and Phononic Crystals. Crystals 2020, 10, 305. [CrossRef] 26. Bertoldi, K.; Vitelli, V.; Christensen, J.; van Hecke, M. Flexible Mechanical Metamaterials. Nat. Rev. Mater. 2017, 2, 1–11. [CrossRef] 27. Yang, H.; Ma, L. Multi-Stable Mechanical Metamaterials by Elastic Buckling Instability. J. Mater. Sci. 2019, 54, 3509–3526. [CrossRef] 28. Kochmann, D.M.; Bertoldi, K. Exploiting Microstructural Instabilities in Solids and Structures: From Metamaterials to Structural Transitions. Appl. Mech. Rev. 2017, 69.[CrossRef] 29. Slesarenko, V. Planar Mechanical Metamaterials with Embedded Permanent Magnets. Materials 2020, 13, 1313. [CrossRef] 30. Shim, J.; Shan, S.; Košmrlj, A.; Kang, S.H.; Chen, E.R.; Weaver, J.C.; Bertoldi, K. Harnessing Instabilities for Design of Soft Reconﬁgurable Auxetic/Chiral Materials. Soft Matter 2013, 9, 8198–8202. [CrossRef] 31. Ghaedizadeh, A.; Shen, J.; Ren, X.; Xie, Y.M. Tuning the Performance of Metallic Auxetic Metamaterials by Using Buckling and Plasticity. Materials 2016, 9, 54. [CrossRef][PubMed] 32. Li, J.; Slesarenko, V.; Rudykh, S. Auxetic Multiphase Soft Composite Material Design through Instabilities with Application for Acoustic Metamaterials. Soft Matter 2018, 14, 6171–6180. [CrossRef][PubMed] 33. Shim, J.; Wang, P.; Bertoldi, K. Harnessing Instability-Induced Pattern Transformation to Design Tunable Phononic Crystals. Int. J. Solids Struct. 2015, 58, 52–61. [CrossRef] 34. Slesarenko, V.; Galich, P.I.; Li, J.; Fang, N.X.; Rudykh, S. Foreshadowing Elastic Instabilities by Negative Group Velocity in Soft Composites. Appl. Phys. Lett. 2018, 113.[CrossRef] 35. Li, J.; Arora, N.; Rudykh, S. Elastic Instabilities, Microstructure Transformations, and Pattern Formations in Soft Materials. Curr. Opin. Solid State Mater. Sci. 2021, 25, 100898. [CrossRef] 36. Goshkoderia, A.; Chen, V.; Li, J.; Juhl, A.; Buskohl, P.; Rudykh, S. Instability-Induced Pattern Formations in Soft Magnetoactive Composites. Phys. Rev. Lett. 2020, 124, 158002. [CrossRef] 37. Li, J.; Slesarenko, V.; Rudykh, S. Microscopic Instabilities and Elastic Wave Propagation in Finitely Deformed Laminates with Compressible Hyperelastic Phases. Eur. J. Mech. A Solids 2019, 73, 126–136. [CrossRef] 38. Arora, N.; Li, J.; Slesarenko, V.; Rudykh, S. Microscopic and Long-Wave Instabilities in 3D Fiber Composites with Non-Gaussian Hyperelastic Phases. Int. J. Eng. Sci. 2020, 157.[CrossRef] 39. Li, J.; Slesarenko, V.; Galich, P.I.; Rudykh, S. Instabilities and Pattern Formations in 3D-Printed Deformable Fiber Composites. Compos. Part B Eng. 2018, 148, 114–122. [CrossRef] 40. Slesarenko, V.; Rudykh, S. Microscopic and Macroscopic Instabilities in Hyperelastic Fiber Composites. J. Mech. Phys. Solids 2017, 99, 471–482. [CrossRef] 41. Triantafyllidis, N.; Maker, B.N. On the Comparison Between Microscopic and Macroscopic Instability Mechanisms in a Class of Fiber-Reinforced Composites. J. Appl. Mech. 1985, 52, 794. [CrossRef] 42. Geymonat, G.; Müller, S.; Triantafyllidis, N. Homogenization of Nonlinearly Elastic Materials, Microscopic Bifurcation and Macroscopic Loss of Rank-One Convexity. Arch. Ration. Mech. Anal. 1993, 122, 231–290. [CrossRef] 43. Merodio, J.; Ogden, R.W. Material Instabilities in Fiber-Reinforced Nonlinearly Elasti Solids under Plane Deformation. Arch. Mech. 2002, 54, 525–552. 44. Merodio, J.; Ogden, R.W. On Tensile Instabilities and Ellipticity Loss in Fiber-Reinforced Incompressible Non-Linearly Elastic Solids. Mech. Res. Commun. 2005, 32, 290–299. [CrossRef] 45. Merodio, J.; Ogden, R.W. Remarks on Instabilities and Ellipticity for a Fiber-Reinforced Compressible Nonlinearly Elastic Solid under Plane Deformation. Q. Appl. Mathermatics 2005, LXIII, 325–333. [CrossRef] 46. DeBotton, G. Transversely Isotropic Sequentially Laminated Composites in Finite Elasticity. J. Mech. Phys. Solids 2005, 53, 1334–1361. [CrossRef] 47. Agoras, M.; Lopez-Pamies, O.; Ponte Castañeda, P. Onset of Macroscopic Instabilities in Fiber-Reinforced Elastomers at Finite Strain. J. Mech. Phys. Solids 2009, 57, 1828–1850. [CrossRef] 48. Rudykh, S.; DeBotton, G. Instabilities of Hyperelastic Fiber Composites: Micromechanical Versus Numerical Analyses. J. Elast. 2012, 106, 123–147. [CrossRef] 49. Nestorovi´c,M.D.D.; Triantafyllidis, N. Onset of Failure in Finitely Strained Layered Composites Subjected to Combined Normal and Shear Loading. J. Mech. Phys. Solids 2004, 52, 941–974. [CrossRef] 50. Liu, J.; Bertoldi, K. Bloch Wave Approach for the Analysis of Sequential Bifurcations in Bilayer Structures. Proc. R. Soc. Math. Phys. Eng. Sci. 2015, 471, 20150493. [CrossRef] 51. Cutolo, A.; Pagliarulo, V.; Merola, F.; Coppola, S.; Ferraro, P.; Fraldi, M. Wrinkling Prediction, Formation and Evolution in Thin Films Adhering on Polymeric Substrata. Mater. Des. 2020, 187, 108314. [CrossRef] 52. Yin, J.; Yagüe, J.L.; Eggenspieler, D.; Gleason, K.K.; Boyce, M.C. Deterministic Order in Surface Micro-Topologies through Sequential Wrinkling. Adv. Mater. 2012, 24, 5441–5446. [CrossRef] Materials 2021, 14, 2038 13 of 13

53. Zheng, Y.; Li, G.-Y.; Cao, Y.; Feng, X.-Q. Wrinkling of a Stiff Film Resting on a Fiber-Filled Soft Substrate and Its Potential Application as Tunable Metamaterials. Extreme Mech. Lett. 2017, 11, 121–127. [CrossRef] 54. Zhang, T. Symplectic Analysis for Wrinkles: A Case Study of Layered Neo-Hookean Structures. J. Appl. Mech. 2017, 84.[CrossRef] 55. Xie, W.-H.; Huang, X.; Cao, Y.-P.; Li, B.; Feng, X.-Q. Buckling and Postbuckling of Stiff Lamellae in a Compliant Matrix. Compos. Sci. Technol. 2014, 99, 89–95. [CrossRef] 56. Coulais, C.; Sabbadini, A.; Vink, F.; van Hecke, M. Multi-Step Self-Guided Pathways for Shape-Changing Metamaterials. Nature 2018, 561, 512–515. [CrossRef][PubMed] 57. Gao, C.; Slesarenko, V.; Boyce, M.C.; Rudykh, S.; Li, Y. Instability-Induced Pattern Transformation in Soft Metamaterial with Hexagonal Networks for Tunable Wave Propagation. Sci. Rep. 2018, 8, 11834. [CrossRef][PubMed] 58. Nassar, H.; Chen, H.; Huang, G. Microtwist Elasticity: A Continuum Approach to Zero Modes and Topological Polarization in Kagome Lattices. J. Mech. Phys. Solids 2020, 144, 104107. [CrossRef] 59. Peraza-Hernandez, E.A.; Hartl, D.J.; Malak, R.J., Jr.; Lagoudas, D.C. Origami-Inspired Active Structures: A Synthesis and Review. Smart Mater. Struct. 2014, 23, 094001. [CrossRef] 60. Xu, X.; Wang, C.; Shou, W.; Du, Z.; Chen, Y.; Li, B.; Matusik, W.; Hussein, N.; Huang, G. Physical Realization of Elastic Cloaking with a Polar Material. Phys. Rev. Lett. 2020, 124, 114301. [CrossRef] 61. Aberg, M.; Gudmundson, P. The Usage of Standard Finite Element Codes for Computation of Dispersion Relations in Materials with Periodic Microstructure. J. Acoust. Soc. Am. 1997, 102, 2007–2013. [CrossRef] 62. Bertoldi, K.; Boyce, M.C. Wave Propagation and Instabilities in Monolithic and Periodically Structured Elastomeric Materials Undergoing Large Deformations. Phys. Rev. B 2008, 78, 184107. [CrossRef] 63. He, Y.; Zhou, Y.; Liu, Z.; Liew, K.M. Buckling and Pattern Transformation of Modiﬁed Periodic Lattice Structures. Extreme Mech. Lett. 2018, 11. [CrossRef] 64. Galich, P.I.; Thomas, E. Soft Modes in Nonlinear Composites on the Edge of Elastic Instability. In Proceedings of the 26th International Congress on Sound and Vibration, Montreal, QC, Canada, 7–11 July 2019. 65. Galich, P.I.; Slesarenko, V.; Rudykh, S. Shear Wave Propagation in Finitely Deformed 3D Fiber-Reinforced Composites. Int. J. Solids Struct. 2017, 110–111, 294–304. [CrossRef] 66. Maurin, F.; Claeys, C.; Deckers, E.; Desmet, W. Probability That a Band-Gap Extremum Is Located on the Irreducible Brillouin-Zone Contour for the 17 Different Plane Crystallographic Lattices. Int. J. Solids Struct. 2018, 135, 26–36. [CrossRef] 67. Li, J.; Pallicity, T.D.; Slesarenko, V.; Goshkoderia, A.; Rudykh, S. Domain Formations and Pattern Transitions via Instabilities in Soft Heterogeneous Materials. Adv. Mater. 2019, 31.[CrossRef] 68. Zorzetto, L.; Andena, L.; Briatico-Vangosa, F.; De Noni, L.; Thomassin, J.-M.; Jérôme, C.; Grossman, Q.; Mertens, A.; Weinkamer, R.; Rink, M.; et al. Properties and Role of Interfaces in Multimaterial 3D Printed Composites. Sci. Rep. 2020, 10, 22285. [CrossRef] 69. Arora, N.; Batan, A.; Li, J.; Slesarenko, V.; Rudykh, S. On the Inﬂuence of Inhomogeneous Interphase Layers on Instabilities in Hyperelastic Composites. Materials 2019, 12, 763. [CrossRef]