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Soft cellular structures that comprise a solid matrix with a square array of holes open avenues for the design of novel soft and foldable structures. Our results demonstrate that by simply changing the shape of the holes the response of porous structure can be easily tuned and soft structures with optimal compaction can be designed.

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... Unlike the NS mechanisms that have already been applied in industrial fields including micro-electro-mechanical systems (MEMS) [4], vibration isolation [5], and amplifying dampers [6], the NS mechanical metamaterial [7], comprising a periodic arrangement of NS unit cells with artificial microstructures is an emerging realm and started to gain increasing attention in the last decade. It has been demonstrated to have great prospects in energy absorption , vibration isolation [26,[48][49][50][51][52], advanced actuators [50], and deployable structures [51,52] at both macroscale [12][13][14][17][18][19][20][21][22][23][24][27][28][29][30][31][32]34,36,[43][44][45][46][47] and microscale [15,16,25,26,33,[37][38][39][40][41][42], which exhibits the similar characteristics as some artificial auxetic structures [53][54][55]. Additionally, some examples are capable of performing negative Poisson's ratio and negative stiffness at the same time [31,32,35,36,44]. ...

... It has been demonstrated to have great prospects in energy absorption , vibration isolation [26,[48][49][50][51][52], advanced actuators [50], and deployable structures [51,52] at both macroscale [12][13][14][17][18][19][20][21][22][23][24][27][28][29][30][31][32]34,36,[43][44][45][46][47] and microscale [15,16,25,26,33,[37][38][39][40][41][42], which exhibits the similar characteristics as some artificial auxetic structures [53][54][55]. Additionally, some examples are capable of performing negative Poisson's ratio and negative stiffness at the same time [31,32,35,36,44]. ...

... Despite the prefabricated curved beam, other micro-structures can also be introduced as a functional component that can generate the BNS effect. For instance, Overvelde et al. [31] and Florijn et al. [32] designed different 2D periodic porous metamaterials and investigated the buckling effect of pore shapes and constraints on their mechanical properties. Meza et al. [33] designed a lightweight and high-strength BNS metamaterial with thin-walled tubes that produce recoverable local buckling. ...

Negative stiffness (NS) metamaterials are kinds of artificial mechanical materials that exhibit the negative stiffness effect governed by different mechanical mechanisms, among which the buckling-based negative stiffness (BNS) metamaterial consisting of prefabricated curved beams and rigid frames have attracted considerable concern. However, the variety of the reported geometrical configurations of BNS metamaterials is relatively limited, and the study of systematically developing multidirectional buckling-based negative stiffness (MDBNS) metamaterials has rarely been seen at present, especially their mechanical performance. Therefore, the novelty of this study is to design a systematic group of MDBNS metamaterials comprising flexible cosine curved beams and stiffened supporting frames based on Bravais lattices. The group contains 15 element members and has been classified into 5 categories including the 2D bi-directional, 2D tri-directional, 3D tri-directional, 3D quadri-directional, and 3D six-directional NS functional elements. The geometrical configurations of the proposed MDBNS functional elements are respectively constructed based on the systematic arrangement rule of the Bravais lattices that provides the design guidance for the systematic metamaterial group with a fixed number of group members. The mechanical performance of some typical functional elements with the most NS directions and the metamaterial lattices composed of the proposed MDBNS metamaterials have been verified via the Finite Element Analysis (FEA) with the periodic boundary condition (PBC), respectively, while the numerical study has been validated experimentally with 2D MDBNS functional elements via 3D additive manufacturing. The results strongly prove the mechanical NS effects of the proposed MDBNS metamaterials and also indicate that the mechanical behavior is independent if the NS directions are orthogonal, whereas it interacts with each other in the case of non-orthogonal NS directions. The study can also be adopted to develop multidirectional NS metamaterials with other NS functional components and induced mechanisms.

... Previously, most SMP structures were trained by global pre-strains from mold and external mechanical loads 46,47 . Our deformation capability with local magnetic torque can reconfigure with complex geometry such as functionally graded structures in Figure 2b, which could not build with the traditional training method of SMPs 17,31,32 . ...

... Our deformation capability with local magnetic torque can reconfigure with complex geometry such as functionally graded structures in Figure 2b, which could not build with the traditional training method of SMPs 17,31,32 . It is also highly challenging to hold a buckled shape of soft structures without releasing external loadings 46,47 ; however, the shape fixity of SMPs can resolve this issue in this work. Moreover, it is challenging to generate a higher mode (a low-wavelength mode) deformation of unstable structures 46,47 . ...

... It is also highly challenging to hold a buckled shape of soft structures without releasing external loadings 46,47 ; however, the shape fixity of SMPs can resolve this issue in this work. Moreover, it is challenging to generate a higher mode (a low-wavelength mode) deformation of unstable structures 46,47 . Our asymmetric magnetization arrangement and instability design can generate a higher mode deformation, as demonstrated in Figure 4a. ...

Future active metamaterials for reconfigurable structural applications require fast, untethered, reversible, and reprogrammable (multimodal) transformability with shape locking. Herein, we aim to construct and demonstrate a magneto-thermomechanical tool that enables a single material system to transform with untethered, reversible, low-powered reprogrammable deformations and shape locking via the application of magneto-thermomechanically triggered prestress on a shape memory polymer and structural instability with asymmetric magnetic torque. We demonstrate the mutual assistance of two physics concepts - magnetic control combined with the thermomechanical behavior of shape memory polymers, without requiring new materials synthesis and high-power energy for reprogramming. Our approach can open a new path of active metamaterials, flexible yet stiff soft robots, and multimodal morphing structures, where we can design them in reversible and reprogrammable ways.

... Negative Poisson's ratio materials were founded in the 1980s [22][23][24][25]; since then, considerable efforts have been devoted designing, modeling, and analyzing auxetic metamaterials [26][27][28][29]. Despite such intensive efforts to design new auxetic metamaterials, it remains challenging to design flexible auxetic metamaterials with extreme properties, such as materials that can maintain negative Poisson's ratios consistently during large deformations [21,39,40,32]. For example, some delicately designed structures have positive Poisson's ratios under small deformations and only exhibit negative Poisson's ratios during further compression [39,40,32]. ...

... Despite such intensive efforts to design new auxetic metamaterials, it remains challenging to design flexible auxetic metamaterials with extreme properties, such as materials that can maintain negative Poisson's ratios consistently during large deformations [21,39,40,32]. For example, some delicately designed structures have positive Poisson's ratios under small deformations and only exhibit negative Poisson's ratios during further compression [39,40,32]. This is because of the limitations of conventional design methods. ...

... This is because of the limitations of conventional design methods. Forward design is the mainstream method for designing auxetic metamaterials and includes approaches such as bioinspired methods [30,31], mathematical control [32][33][34][35], topology optimization [36][37][38], and Boolean and lofting operations of simple geometries [39][40][41][42][43]. The forward design approach follows a general process: first, a structure is created, and its mechanical properties are then investigated by finite element method (FEM) simulations or mechanical testing (Fig. 1). ...

As typical mechanical metamaterials with negative Poisson’s ratios, auxetic metamaterials exhibit counterintuitive auxetic behaviors that are highly dependent on their geometric arrangements. The realization of the geometric arrangement required to achieve a negative Poisson’s ratio relies considerably on the experience of designers and trial-and-error approaches. This report proposes an inverse design method for auxetic metamaterials using deep learning, in which a batch of auxetic metamaterials with a user-defined Poisson’s ratio and Young’s modulus can be generated by a conditional generative adversarial network without prior knowledge. The network was trained based on supervised learning using a large number of geometrical patterns generated by Voronoi tessellation. The performance of the network was demonstrated by verifying the mechanical properties of the generated patterns using finite element method simulations and uniaxial compression tests. The successful realization of user-desired properties can potentially accelerate the inverse design and development of mechanical metamaterials.

... Mechanical metamaterials (MMs) are commonly used in engineering for tailoring unique mechanical properties as their mechanics are primarily governed by their geometry rather than compositions. Recent development of fabrication technology such as additive manufacture has stimulated the use of soft MMs to realize extreme mechanical properties such as negative Poisson's ratio [Bertoldi et al., 2010, Overvelde et al., 2012, Overvelde and Bertoldi, 2014, shape morphing [Mirzaali et al., 2018], tunable band structures [Krishnan and Johnson, 2009], energy absorption [Meza et al., 2014]. These unique properties opened up the possibility of many exciting engineering applications, e.g., soft actuators, materials with in situ tunable functionalities, reusable energy-absorbing materials [Li and Gao, 2016, Florijn et al., 2016, Bertoldi, 2017, Barchiesi et al., 2019, Surjadi et al., 2019, etc. ...

... In this work, we constrain ourselves to the study of 2D soft cellular mechanical metamaterials (CMMs) made of square unit-cells with a pore in the center. Inspired by previous works [Overvelde et al., 2012, Overvelde and, we consider the pore shapes with four-fold symmetry whose contour can be described by the following equation: ...

... Previous studies have shown that the mechanical properties of CMMs are highly sensitive to the pore shapes [Bertoldi et al., 2010, Overvelde et al., 2012, Overvelde and Bertoldi, 2014, Xue et al., 2020. Under uniaxial compression, mechanical instabilities at the scale of unit-cell often lead to pattern transformations at the structural scale. ...

The dynamics of soft mechanical metamaterials provides opportunities for many exciting engineering applications. Previous studies often use discrete systems, composed of rigid elements and nonlinear springs, to model the nonlinear dynamic responses of the continuum metamaterials. Yet it remains a challenge to accurately construct such systems based on the geometry of the building blocks of the metamaterial. In this work, we propose a machine learning approach to address this challenge. A metamaterial graph network (MGN) is used to represent the discrete system, where the nodal features contain the positions and orientations the rigid elements, and the edge update functions describe the mechanics of the nonlinear springs. We use Gaussian process regression as the surrogate model to characterize the elastic energy of the nonlinear springs as a function of the relative positions and orientations of the connected rigid elements. The optimal model can be obtained by "learning" from the data generated via finite element calculation over the corresponding building block of the continuum metamaterial. Then, we deploy the optimal model to the network so that the dynamics of the metamaterial at the structural scale can be studied. We verify the accuracy of our machine learning approach against several representative numerical examples. In these examples, the proposed approach can significantly reduce the computational cost when compared to direct numerical simulation while reaching comparable accuracy. Moreover, defects and spatial inhomogeneities can be easily incorporated into our approach, which can be useful for the rational design of soft mechanical metamaterials.

... i.e., tubes placed at regular intervals. 43,49 Since shape-memory polymeric aerogels are superelastic to begin with, it was reasoned that fabrication of such auxetic metastructures with polymeric aerogels that show the shapememory effect would comprise a new class of materials, referred to herewith as meta-aerogels, which will be especially suitable as deployable panels in aerospace applications as they will contract and occupy less space during stowing in their folded temporary shape on their way to the location of their deployment in their permanent shape. ...

... 45,50 Recently, Bertoldi et al. studied the effect of the shape of the cross-section of the tubes on the compressive behavior of such 2D periodic structures and showed that the negative Poisson's ratios through buckling instabilities do depend on the cross-sectional shape of the tubes. 49 The crosssectional shape of the tubes chosen for this work ( Figure 1A − inset) was selected as the one with the most negative Poisson's ratio from the shapes investigated by Betroldi. 49 That auxetic structure was reconstructed using the SolidWorks 3D CAD software, 51 using a parametric function in polar coordinates with the center-to-center distance between neighboring holes set at L 0 = 10 mm; the total porosity due to the tubes was set at φ = 0.47, and the number of repeat unit cells within the structure was set at N = 8. ...

... 49 The crosssectional shape of the tubes chosen for this work ( Figure 1A − inset) was selected as the one with the most negative Poisson's ratio from the shapes investigated by Betroldi. 49 That auxetic structure was reconstructed using the SolidWorks 3D CAD software, 51 using a parametric function in polar coordinates with the center-to-center distance between neighboring holes set at L 0 = 10 mm; the total porosity due to the tubes was set at φ = 0.47, and the number of repeat unit cells within the structure was set at N = 8. Other details for the design are given in Figure S.1 of Appendix I in the Supporting Information. ...

Shape-memory poly(isocyanurate−urethane) (PIR−PUR) aerogels are low-density monolithic nanoporous solids that remember and return to their permanent shape through a heating actuation step. Herein, through structural design at the macro scale, the shape-memory response is augmented with an auxetic effect manifested by a negative Poisson’s ratio of approximately −0.8 at 15% compressive strain. Thus, auxetic shape-memory PIR−PUR monoliths experience volume contraction upon compression at a temperature above the glass transition temperature of the base polymer (Tg ≈ 30 °C), and they can be stowed indefinitely in that temporary shape by cooling below Tg. By heating back above Tg, the compressed/shrunk form expands back to their original shape/size. This technology is relevant to a broad range of industries spanning the commercial, aeronautical, and aerospace sectors. The materials are referred to as meta-aerogels, and their potential applications include minimally invasive medical devices, soft robotics, and situations where volume is at a premium, as for example for storage of deployable space structures and planetary habitats during transport to the point of service.

... Buckling, in particular, is an elastic instability occurring at a critical stress with large displacements, without any relevant addition of stress, and has been recently highlighted as a novel actuation mechanism for soft robots due to the potential for control in the deformation obtained. This instability can be used to produce adaptive structures with controlled shape changes, allowing for the design of responsive and reconfigurable devices [22,28]. For instance, two-dimensional periodic elastomeric porous structures undergo controlled deformation with the critical buckling stress, transforming a square array of circular holes into periodic patterns of alternating orthogonal ellipses through a cooperative torsion and collapse of the beams [28][29]. ...

... This instability can be used to produce adaptive structures with controlled shape changes, allowing for the design of responsive and reconfigurable devices [22,28]. For instance, two-dimensional periodic elastomeric porous structures undergo controlled deformation with the critical buckling stress, transforming a square array of circular holes into periodic patterns of alternating orthogonal ellipses through a cooperative torsion and collapse of the beams [28][29]. These geometries and patterns facilitate the design of a family of buckling-driven mechanical metamaterials, thanks to their properties of high level programmability and negative Poisson ratios (auxetic behavior); these properties are useful to program the deformation of the structures [30]. ...

... Another relevant result is related to the configuration stability. Current elastic buckling instabilities can jump to a set of different final configurations [28,50], as observed in the squared structures and the single circular chamber, showing bistability and, therefore, two random final shapes (see inserts in Fig. 3). This random postbuckling behavior arises from the asymmetric bifurcation response, preventing its application in actuators and energy harvesting [53]. ...

Mechanical instabilities are emerging as novel actuation mechanisms for the design of biomimetic soft robots and smart structures. The present study shows that by coupling buckling-driven elastomeric auxetic modules actuated by a negative air-pressure, a novel metamaterial-based caterpillar can be designed-the Metarpillar. Following a detailed analysis of the caterpillar's locomotion, we were able to mimic both its crawling movement and locomotion by using the unique isometric compression of the modules and properly programing the anterograde modular peristaltic contractions. The bioinspired locomotion of the Metarpillar uses the bending triggered by the buckling-driven module contraction to control the friction through a dynamic anchoring between the soft robot and the surface, which is the main mechanism for locomotion in caterpillars and other crawling organisms. Thus, the Metarpillar not only mimics the locomotion of the caterpillar but also displays dynamic similarity and equivalent, or even faster, speeds. Our approach based on metamaterial buckling actuator units opens up a novel strategy for biomimetic soft robotic locomotion that can be extended beyond caterpillars.

... Similar to these crystallographic networks, the properties of periodic cellular structures can be designed by controlling the architecture of the repeating unit cell (3). This results in the purposeful design of cellular structures with advanced macroscopic mechanical properties such as negative Poisson's ratios (4, 5), excellent strength-toweight ratios (6, 7), and controlled instabilities (8,9). While finding novel microstructures has proven essential for the development of future applications and technologies (10, 11), many architected cellular structures in engineering (12, 13), fundamental science (14-16), and advanced manufacturing (17, 18) are based on a small selection of well-studied designs. ...

... Similar to these crystallographic networks, the properties of periodic cellular structures can be designed by controlling the architecture of the repeating unit cell (3). This results in the purposeful design of cellular structures with advanced macroscopic mechanical properties such as negative Poisson's ratios (4,5), excellent strength-toweight ratios (6,7), and controlled instabilities (8,9). While finding novel microstructures has proven essential for the development of future applications and technologies (10,11), many architected cellular structures in engineering (12,13), fundamental science (14)(15)(16), and advanced manufacturing (17,18) are based on a small selection of well-studied designs. ...

Significance
Finding genuine novelty in cellular structures is inherently difficult due to the numerous possible topological and geometrical configurations and their complex mechanical and physical interrelations. Here, we draw inspiration from the incredibly rich collection of crystallographic periodic networks that we interpret from a structural point of view to identify and design novel cellular structures with unique properties. We provide a ready-to-use catalog with more than 17,000 unique entries and show how crystallographic symmetries relate to their mechanical properties. Our work provides a foundation to support future applications in science and engineering, ranging from mechanical and optical metamaterials, over bone tissue engineering, to the design of electrochemical devices.

... System and procedure.-We design a hollow cylindrical shell composed of an array of unit-cells, which provides a network of non-uniform beams (FIG 2a) capable of side-buckling and self-contacting under compression [19]. The cylindrical shell has an outer radius of R max = 12.5mm, and an inner radius of R min = 7.5mm. ...

... and b = 0.28 (FIG 1 SM. a), exhibits side-buckling under compression [19]. This shape also provides a nonuniform profile for the cross-section of the beams that form the structure. ...

The Poynting effect generically manifests itself as the extension of the material in the direction perpendicular to an applied shear deformation (torsion) and is a material parameter hard to design. Here, we engineer a metamaterial that can be programmed to contract or extend under torsion, depending on its architecture. First, we show that our system exhibits a novel type of inverted Poynting effect, where axial compression induces a nonlinear torsion. Then we program the Poynting modulus of the structure from initial negative values to zero and positive values via a pre-compression applied prior to torsion. Our work opens avenues for programming nonlinear elastic response of materials by rational design.

... The bending and buckling behavior of a porous plate subjected to uniformly distributed force and buckling force was studied by Magnucka-Blandzi [20] and the critical load linearly decreased with the increased porosity of the plate. Overvelde et al. investigated two-dimensional soft porous lattice for the influence of structure shapes on buckling and reported that the structure's design affects the buckling actions of the soft, porous system [21]. Saghaian et al. [22] systematically investigated three various TPMS designs with constant porosity levels and found that the mechanical properties of porous samples were highly dependent on the structure's geometry. ...

... This failure is caused by the compression of the column [40]. The critical buckling load of lattice structures depends on various parameters, such as the length of the column, wall thickness, relative density, tessellation, the presence of the column inside the structure and lattice pore size [21,[41][42][43][44]. Among the designs employed in this research, Diamond samples exhibit a consistent drop in critical buckling load with increasing height; hence, the Diamond design may be used for columns when height is necessary. ...

Additive Manufacturing (AM) is rapidly evolving due to its unlimited design freedom to fabricate complex and intricate light-weight geometries with the use of lattice structure that have potential applications including construction, aerospace and biomedical applications, where mechanical properties are the prime focus. Buckling instability in lattice structures is one of the main failure mechanisms that can lead to major failure in structural applications that are subjected to compressive loads, but it has yet to be fully explored. This study aims to investigate the effect of surface-based lattice structure topologies and structured column height on the critical buckling load of lattice structured columns. Four different triply periodic minimal surface (TPMS) lattice topologies were selected and three design configurations (unit cells in x, y, z axis), i.e., 2 × 2 × 4, 2 × 2 × 8 and 2 × 2 × 16 column, for each structure were designed followed by printing using HP MultiJet fusion. Uni-axial compression testing was performed to study the variation in critical buckling load due to change in unit cell topology and column height. The results revealed that the structured column possessing Diamond structures shows the highest critical buckling load followed by Neovius and Gyroid structures, whereas the Schwarz-P unit cell showed least resistance to buckling among the unit cells analyzed in this study. In addition to that, the Diamond design showed a uniform decrease in critical buckling load with a column height maximum of 5193 N, which makes it better for applications in which the column’s height is relatively higher while the Schwarz-P design showed advantages for low height column maximum of 2271 N. Overall, the variations of unit cell morphologies greatly affect the critical buckling load and permits the researchers to select different lattice structures for various applications as per load/stiffness requirement with different height and dimensions. Experimental results were validated by finite element analysis (FEA), which showed same patterns of buckling while the numerical values of critical buckling load show the variation to be up to 10%.

... Since then, diverse studies were dedicated to understanding the influences of the loading scenarios, material nonlinearities, pore arrangements and geometries, porosities, pattern transitions, etc. on the pattern formation and the accompanied bandgaps in soft metamaterials. Overvelde et al. [24] discussed the role of pore shape in controlling the compaction performance through buckling based on both experimental and numerical results and clarified that the circular hole is not an optimal shape. Wang et al. [25] revealed the effects of geometrical and material nonlinearities, as well as the applied strain on the evolution of bandgaps by considering an equi-biaxial compression and by employing the neo-Hookean and Gent material constitutions. ...

... where l b signifies the effective beam length and can be evaluated by seeking the boundary between blue and grey meshes (blue denotes a nearly zero deformation). Solving equation (24) subject to the boundary condition (25) yields Subsequently, we plan to determine the correction factor α in (23) and to check the validity of the proposed theoretical models. To this end, we fix T = 0.1 mm from now on and alter the hole radius from A = 5.6 mm to A = 6.5 mm. ...

This paper proposes a new metamaterial structure consisting of a periodically porous elastomer with pore coatings. This design enables us to engender finite deformation by a contactless load. As a case study, we apply thermal load to the pore coating and carry out a finite element analysis to probe instabilities and the associated phononic properties. It turns out that a novel buckling mode, preserving the nature of surface wrinkling in tubular structures, can be induced under a plane-strain setup, and a smaller size of the unit cell is attained compared to the counterpart of traditional buckled profile in soft porous elastomers. In particular, this buckling pattern is able to produce several bandgaps in different frequency ranges as the macroscopic mean strain increases. We further introduce a metallic core as local resonator, and the updated metamaterial allows a low-frequency bandgap, the bandgap width of which can be estimated by a simplified theoretical model. As more free parameters are involved in the structure, we perform a detailed parametric study to elucidate the influences of the modulus ratio between coating and matrix, the porosity, the core radius, and the macroscopic mean strain on the buckling initiation and the evolution of bandgap. Remarkably, a stiffer surface coating is prone to enhance the stability of the structure, which is contrary to existing results in film/substrate bilayers. It is expected that the current study could shed light on new insight into pattern formation and wave manipulation in porous elastomers.

... Mechanical behavior of transversely isotropic porous Neo-Hookean solids under shear loading is studied [Guo and Caner, 2010]. Plenty of work has shown that the buckling mode of periodic porous members can be tuned by some external parameters such as the inclusion of the holes, geometry of primitive cells and loading condition Overvelde et al., 2012;. Though there is still a lack of study on the variation of buckling pattern transition points influenced by the change of multiaxial loading proportions and representative volume element (RVE) size. ...

... Comparing with the previous works [Overvelde et al., 2012;, multiaxial loading condition can be implemented through the PBC model proposed here. Two loading parameters, Φ 1 and Φ 2 , are introduced to control the prescribed proportional loading state. ...

This paper focuses on the buckling instabilities of periodic porous elastomers under combined multiaxial loading. A numerical model based on the periodic boundary condition (PBC) for the 2D representative volume element (RVE) is proposed, in which two proportional loading parameters are employed to control the complex stressing state applied to the RVE model. A homogenization-based orthogonal transformation matrix is established by satisfying the equality of the total work rate to realize a proportional multiaxial loading on the RVE. First, the transition behavior of buckling patterns of periodic porous structures is revealed through instability analysis for the RVE consisting of [Formula: see text] primitive cells with circular holes subjected to different proportional loading conditions. Simulation results show that the first-order buckling mode of RVE may change suddenly from a uniaxial shearing buckling pattern to a biaxial rotating buckling pattern at a critical loading proportion. Then the influences of the number of primitive cells in the enlarged RVE on the buckling behavior are discussed. When the number of primitive cells in any enlarging direction is odd, the points of buckling pattern transition of the enlarged RVEs vary significantly with the number of cells in RVE. When the number of primitive cells is even in both enlarging directions, there is no apparent difference for the critical buckling stresses of the enlarged RVEs.

... This was both observed during experimental tests and retraced via theoretical models and numerical simulations, which revealed how the originally circular pores deform into elliptic shapes with alternating orthogonal directions of the major axes. As a result, such kind of kinematics leads to have, for the equivalent macroscopic body, a negative Poisson's ratio associated to a significant shape compaction [41] and to an elastic plateau-like trend in the stress-strain response [18,40]. On this basis, topology optimization of a porous elastomeric matrix was also performed by Overvelde et al. [41] with the aim to investigate the influence of the holes' shape on the nonlinear response of the equivalent medium and thus provide a way to control and improve desired mechanical performances. ...

... As a result, such kind of kinematics leads to have, for the equivalent macroscopic body, a negative Poisson's ratio associated to a significant shape compaction [41] and to an elastic plateau-like trend in the stress-strain response [18,40]. On this basis, topology optimization of a porous elastomeric matrix was also performed by Overvelde et al. [41] with the aim to investigate the influence of the holes' shape on the nonlinear response of the equivalent medium and thus provide a way to control and improve desired mechanical performances. On the other hand, the effects of a triangular packing of circular holes within soft media were studied by Shan et al. [42], thus highlighting, via laboratory tests and Finite Element (FE) analyses, the possibility to induce multiple pattern transformations by changing the direction of the prescribed loads and to accordingly tune the dynamic response, in particular the band-gaps of the overall system. ...

With in mind microstructures exhibiting unconventional macroscopic mechanical behaviors, characterized by overall auxetic responses and strain localization due to local elastic instabilities, in this work we conceived a simple two-dimensional non-chiral architecture in the form of a periodic lattice, whose drawing is decided by varying the thickness ratios of the cells’ walls and in turn their slenderness. Inspired by nature, which often uses supposedly naive geometries that conceal complex functions, and focusing the attention on two limit geometrical configurations of main interest, we show how these non-chiral settings, as a function of the prescribed boundary conditions, might lead to symmetry breaking associated to a variety of non-trivial deformation modes, which in some cases also retrace and generalize chiral as well as auxetic responses already observed in literature in simpler microstructures. Interestingly, for both the above mentioned limit cases, we were able to idealize the mechanical response of the system through geometrically nonlinear beam-based models, in this way obtaining helpful analytical solutions and explicit formulas to estimate the tangent effective stiffness of the lattice as well as the critical load at the onset of instability. The post-buckling was instead analyzed and discussed in detail through parametric numerical finite element simulations and ad hoc laboratory experiments, performed by faithfully realizing 3D printed prototypes, constructed via additive manufacturing technologies and made of rubber-like material, to follow extreme deformation patterns, including multi-stable states, localization and compaction with possible self-contact/touching of the elements. At the end, we exploited the obtained closed-form solutions and some associated inequalities to derive the transition from the discrete lattice to its continuum limit, which –together with the multiple equilibrium bifurcation points exhibited by the proposed microstructure– could be used to broaden the spectrum of metamaterial geometries designed to exhibit complex tunable properties.

... The interleaved method, on the other hand, uses rigid origami tubes to construct interwoven metamaterials. Numerical approaches have been used for optimal design [173][174][175]197] and systematic study [176] of [189][190][191][192]193,194,195,167,196,197] Cellular metamaterials ...

... Since highly porous materials are assembled by cellular microstructures, the morphology and arrangement of the cellular units (e.g. orientation, size or shape) dominate the mechanical performance of functional materials [163,[186][187][188][189]. Highly porous materials have been used in different industrial applications due to their high internal surface and thermal connectivity, such as heat dissipation, thermal insulation, packaging, or comfortability design [177,230]. ...

Mechanical metamaterials have opened an exciting venue for control and manipulation of architected structures in recent years. Research in the area of mechanical metamaterials has covered many of their fabrication, mechanism characterisation and application aspects. More recently, however, a paradigm shift has emerged to an exciting research direction towards designing, optimising and characterising mechanical metamaterials using artificial intelligence (AI) techniques. This new line of research aims at addressing the difficulties in mechanical metamaterials (i.e. design, analysis, fabrication and industrial application). This review article discusses the advent and development of mechanical metamaterials, and the future trends of applying AI to obtain smart mechanical metamaterials with programmable mechanical response. We explain why architected materials and structures have prominent advantages, what are the main challenges in the mechanical metamaterial research domain, and how to surpass the limit of mechanical metamaterials via the AI techniques. We finally envision the potential research avenues and emerging trends for using the AI-enabled mechanical metamaterials for future innovations.

... Auxetic metamaterials are typically classified into re-entrant [16][17][18][19][20][21][22][23], chiral [24][25][26][27][28], rotating [29][30][31][32][33][34], and hierarchical laminate structures [35][36][37] according to their deformation mechanisms. In general, the re-entrant, chiral, and rotating structures are porous and composed of a single component, whereas the hierarchical laminate structures are solid and consist of two or more components with different Poisson's ratios. ...

An auxetic metamaterial is a type of mechanical metamaterial that has a negative Poisson's ratio. Most auxetic metamaterials are truss-based or originate from Boolean operations of simple geometries. Herein, we introduce a new 3D auxetic metamaterial that is mathematically generated from an implicit expression. Further, this metamaterial is fabricated by 3D printing using a flexible material, which allows it to recover from large deformations. The buckling-induced auxetic behavior of the metamaterial was first evaluated via compression tests and finite element analyses. A nickel layer was then plated onto the surface to enhance its stiffness, strength, and conductivity without loss of auxeticity and resilience. The integration of 3D printing and electroless plating enabled accurate control over the mechanical and conduction properties of the auxetic metamaterial; these properties are presented as contour maps for guidance in functional applications.
We propose a novel 3D auxetic metamaterial derived from a mathematically defined triply periodic minimal surface. The stiffness, strength, and conductivity of the metamaterial are enhanced by nickel plating without loss of auxeticity and resilience. The effective mechanical and conduction properties were mapped against geometric parameters, including relative density and nickel layer thickness. These data maps provide insight for tuning its performance over a broad range.

... The underlying mechanism of the pattern transformations in porous composites is the buckling of thin ligaments between voids, as well as the corresponding folding mechanisms. The initial void shape plays a significant role in the instability induced pattern transformations [91,92] (Fig. 3b); thus, the topological optimization can help achieve the targeted functionalities [92,93]. To rationalize the folding mechanisms in the periodic porous composites, a simplified model has been introduced. ...

Large deformations of soft materials can give rise to the development of various elastic instabilities. The phenomenon is associated with a sudden and dramatic change in structure morphologies. The underlying mechanism is crucial for the formation of complex morphologies in biology. Moreover, the concept of instability-induced pattern transformations is promising for designing novel materials with switchable functions and properties. In this paper, we review the state of the art in elastic instability phenomena in soft materials. We start by considering the classical buckling in beam-based structure lattice designs. Then, we discuss the instability-induced microstructure transformations in soft porous materials, and heterogeneous multiphase and fiber composites. Next, the mechanisms – often involving the post-buckling consideration – leading to the wrinkling and folding, creasing, fringe, and fingering are discussed.

... In the last forty years or so, a variety of methods have been proposed leading to a negative Poisson's ratio. The most prominent methodologies make use of reentrant systems [28][29][30][31][32], involving arrow-shaped ligaments that are pressed in the middle so as to decrease the smaller angle between them; chiral structures [33][34][35][36][37][38][39][40][41][42] in which auxetic behavior is a result of ligament bending and node rotation; rotating rigid or semi-rigid units [43][44][45][46][47], which, as their name implies, achieve this effect through the rotation of units relative to one another; counter rotating units [48][49][50][51] that can harness an internal torque to cause lateral motion; buckling of the material due to elastic instabilities [52][53][54][55][56][57]; and, at a smaller scale, the collective behavior of interacting particles [58][59][60][61][62]. ...

The deformation behavior of intersecting ligaments forming variants of the square and rectangular grids under mechanical compression was investigated. It was shown that such systems are able to exhibit a negative incremental Poisson’s ratio at relatively large axial compressive strains. Numerical simulations and experimental studies indicated that the extent of auxeticity depends on the relative offset of successive ligaments, the relative lengths of the ligaments as well as on their thickness. It was also shown that there are two distinct modes of deformation, one resembling that of the reentrant hexagonal honeycomb and the other that of the meta-tetrachiral system.

... The buckling of lattice structures depends on multi-parameters, such as wall thickness, tessellation, relative density, presence of column inside the structure, lattice pore size, and length of the column [37,[46][47][48][49]. Figure 8 represents the local and global buckling behavior of all uniaxial compressive tested columns. For each type of sample, only one specimen is presented. ...

Lattice structures possess many superior properties over solid materials and conventional structures. Application-oriented lattice structure designs have become a choice in many industries, such as aerospace, automotive applications, construction, biomedical applications, and footwear. However, numerical and empirical analyses are required to predict mechanical behavior under different boundary conditions. In this article, a novel surface-based structure named O-surface structure is designed and inspired by existing Triply Periodic Minimal Surface morphologies in a particular sea urchin structure. For comparison, both structures were designed with two different height configurations and investigated for mechanical performance in terms of compression, local buckling, global buckling, and post-buckling behavior. Both simulation and experimental methods were carried out to reveal these aforementioned properties of samples fabricated by multi jet fusion technology. The sea urchin structure exhibited better mechanical strength than its counterpart, with the same relative density almost two-folds higher in the compressive response. However, the O-surface structure recorded more excellent energy absorption and flexible behavior under compression. Additionally, the compression behavior of the O-surface structure was progressive from top to bottom. In contrast, the sea urchin structure was collapsed randomly due to originated cracks from unit cells’ centers with local buckling effects. Moreover, the buckling direction of structures in long columns was also affected by keeping the relative density constant. Finally, based on specific strength, the O-surface structure exhibited 16-folds higher specific strength than the sea urchin structure.

... Linear and nonlinear effective properties of lattice structures using continuum theory models were proposed. [19,[23][24][25][26] With the advancement in manufacturing technologies like 3D or 4D printing, lattice core with very complex geometrical configurations can be manufactured to amplify the performance of these lightweight sandwich structures. [27] Topology optimization technique was also used to develop new optimal lattice unit cells (ORC, OQSO) which were 5% and 38% stiffer compared to octet lattice unit cell in standard (0 0 1) direction. ...

Structure scouting and design optimization for superior mechanical performance through inverse machine learning is an emerging area of interest. Inverse machine learning can be a substantial approach in structural design to explore complex and massive numbers of geometrical patterns within short periods of time. Here, an inverse design framework using generative adversarial networks (GANs) is proposed to explore and optimize structural designs such as lightweight lattice unit cells. Lightweight lattice structures are widely accepted to have excellent mechanical properties and have found applications in various engineering structures. Using the proposed framework, different lattice unit cells that are 40–120% better in load carrying capacity than octet unit cell are discovered. These new lattice unit cells are analyzed numerically and validated experimentally by testing 3D printed lattice unit cells and lattice cored sandwiches. The proposed inverse design framework can be applied to the design and optimization of other types of load bearing structures.

... Traditionally, columns or rods are optimized in terms of its geometrical shape such as drum-shaped rods have higher buckling load than uniform cylinders 14 . Because the materials around the rod axis do not provide much bending resistance, hollow or porous rods usually have higher buckling resistance than solid rods with the same amount of materials 15,16 . As discussed above, plant stem and root usually have porous structures. ...

Our mother nature has been providing human beings with numerous resources to inspire from, in building a finer life. Particularly in structural design, plenteous notions are being drawn from nature in enhancing the structural capacity as well as the appearance of the structures. Here plant stems, roots and various other structures available in nature that exhibit better buckling resistance are mimicked and modeled by finite element analysis to create a training database. The finite element analysis is validated by uniaxial compression to buckling of 3D printed biomimetic rods using a polymeric ink. After feature identification, forward design and data filtering are conducted by machine learning to optimize the biomimetic rods. The results show that the machine learning designed rods have 150% better buckling resistance than all the rods in the training database, i.e., better than the nature’s counterparts. It is expected that this study opens up a new opportunity to design engineering rods or columns with superior buckling resistance such as in bridges, buildings, and truss structures.

... Nowadays, the auxetic structures have been developed from the original classical double arrows combination structures to chiral tessellation [24] and anti-chiral tessellation structures [25], circular hole structures [26], three-dimensional multi-material structures [27,28], and so on [29]. Baughman pointed out that the phase stability required the sum of all compression for the material was positive, and the material with negative compressibility was incompressible in some directions [30]. ...

Auxetic structure is a typical metamaterial, whose mechanical behaviors of two-dimensional and three-dimensional structures are widely studied. However, reports on surface auxetic structure (SAS) are rare. As a consequence, two types of SAS were designed by reversing and crimping the concave hexagonal plane auxetic structure (PAS) composed of double arrows, which were known as RAS and CAS, respectively. The theoretical equations to calculate the deformation of the representative volume cell structure (RVCS) were derived, and the relationships between energy and work were established based on the plastic wrinkle. The compressive simulations of the plane and surface auxetic structures were conducted through the use of finite element method (FEM) verified by experiment, and the mechanical behaviors and energy absorption characteristics were obtained. By comparing the simulation results of different structures, it was found that RAS not only realized the auxetic effect of compression shrinkage but also realized the supermechanical effect of compression twist. The auxetic effects of these structures were realized by the deformation of beams. The auxetic effect only appeared in the local positions of these structures, and other positions still belonged to the positive Poisson's ratio effect. The crimped CAS had the biggest maximum load peak, and PAS possessed the highest specific energy absorption (SEA). The supermechanical effects of compression shrinkage and compression twist in RAS have great potential in some distinctive engineering applications.

... These materials are usually hyperelastic, such as silicone rubber [19][20][21], polydimethylsiloxane (PDMS) [22,23] or photoelastic elastomer [24,25]. Pattern transformation resulting from the instability opens up a new method for the manufacturing of soft matters with adjustable acoustic, optical and electrical properties [26][27][28][29][30][31]. ...

Structural topology and loading condition have important influences on the mechanical behaviors of porous soft solids. The porous solids are usually set to be under uniaxial tension or compression. Only a few studies have considered the biaxial loads, especially the combined loads of tension and compression. In this study, porous soft solids with oblique and square lattices of circular voids under biaxial loadings were studied through integrated experiments and numerical simulations. For the soft solids with oblique lattices of circular voids, we found a new pattern transformation under biaxial compression, which has alternating elliptic voids with an inclined angle. This kind of pattern transformation is rarely reported under uniaxial compression. Introducing tensile deformation in one direction can hamper this kind of pattern transformation under biaxial loading. For the soft solids with square lattices of voids, the number of voids cannot change their deformation behaviors qualitatively, but quantitatively. In general, our present results demonstrate that void morphology and biaxial loading can be harnessed to tune the pattern transformations of porous soft solids under large deformation. This discovery offers a new avenue for designing the void morphology of soft solids for controlling their deformation patterns under a specific biaxial stress-state.

... As a consequence of this bifurcation pattern, the truss is expected to display auxeticity (Evans, 1991;Reda et al., 2018) in all principle directions at large compressive strains, which we here demonstrate. As described in Overvelde et al. (2012), this concept can be extended to various geometrically more complex beam assemblies. ...

Thanks to scale-bridging fabrication techniques, truss-based metamaterials have gained both popularity and complexity, ultimately resulting in structural networks whose description based on classical discrete numerical calculations becomes intractable. We here present a framework for the efficient and accurate simulation of large periodic three-dimensional (3D) truss networks undergoing nonlinear deformation (accounting for ). Although the focus is on elastic beams, the method is sufficiently general to extend to inelastic material behavior. Our approach is based on a continuum representation of the truss (and its numerical implementation via finite elements) whose constitutive behavior is obtained from on-the-fly periodic homogenization at the microstructural unit cell level. We pursue a semi-analytical strategy (previously reported only in two dimensions) which admits the analytical calculation of consistent tangents for convergent implicit solution schemes; the extension to 3D – through the addition of torsional deformation modes and the handling of 3D rotations – results in a powerful tool for the prediction of the complex mechanical response of large structural networks. We validate the small-strain response by comparison to analytical solutions, followed by finite-strain benchmarks that compare simulation results to those of fully-resolved discrete calculations. The homogenization of beam unit cells results in a regularized macroscale model with an intrinsic length scale, which manifests especially when modeling bifurcations or localization. We finally apply our approach to macroscopic boundary value problems involving complex-shaped truss metamaterials (with truss unit cells near the body’s boundary mapped onto a conformal surface), which reveal only an insignificant effect of boundary layers on the overall mechanical response, again supporting the applicability of our homogenization approach.

... This is also an out-of-plane deformation and a kind of bifurcation phenomenon [32]. This is in stark contrast with the elastomer-based mechanical metamaterials realized by the perforation undergoing in-plane large deformation [33,34]. ...

Nanopapers fabricated from cellulose nanofibers (CNFs) are flexible for bending while they are rather stiff against stretching, which is a common feature shared by conventional paper-based materials in contrast with typical elastomers. Cellulose nanopapers have therefore been expected to be adopted in flexible device applications, but their lack of stretching flexibility can be a bottleneck for specific situations. The high stretching flexibility of nanopapers can effectively be realized by the implementation of Kirigami structures, but there has never been discussion on the mechanical resilience where stretching is not a single event. In this study, we experimentally revealed the mechanical resilience of nanopapers implemented with Kirigami structures for stretching flexibility by iterative tensile tests with large strains. Although the residual strains are found to increase with larger maximum strains and a larger number of stretching cycles, the high mechanical resilience was also confirmed, as expected for moderate maximum strains. Furthermore, we also showed that the round edges of cut patterns instead of bare sharp ones significantly improve the mechanical resilience for harsh stretching conditions. Thus, the design principle of relaxing the stress focusing is not only important in circumventing fractures but also in realizing mechanical resilience.

... In applications involving multistability, the structural elements are usually subjected to large displacements and rotations, and it is often important to consider materials capable of undergoing large deformations without damage. In this scenario, the use of hyperelastic materials finds an important field of application [24,[49][50][51][52][53][54][55]. Also, many applications take advantage of the recent advances in 3D printing [56,57]. ...

Recent decades have witnessed a renewed interest in the field of structural stability due to new applications involving smart and deployable structures, micro- and nanocomponents and mechanical metamaterials, among others. In many of these structures multistable behavior is desirable, which can be accomplished by traditional and new materials capable of undergoing large elastic deformations. In this paper the nonlinear behavior, bifurcations and instabilities of a hyperelastic von Mises truss exhibiting multistable behavior is investigated. Most papers dealing with the von Mises truss are restricted to linear elastic materials. Here, the nonlinear equilibrium equations are derived considering elasticity in the fully non-linear range and the incompressible Mooney–Rivlin constitutive law is adopted to model the hyperelastic material. The nonlinear equations are solved by using the Newton–Raphson method and continuation techniques. Then, all equilibrium paths and bifurcation points are obtained and their stability is investigated using the energy criterion. A detailed parametric analysis of shallow and nonshallow trusses under horizontal and vertical loads is conducted. Load and geometric imperfections are considered and their influence on the bifurcation scenario and the truss load carrying capacity is clarified. The influence of the material parameters on the nonlinear response is also examined. The results show that the simultaneous presence of geometric and material nonlinearities leads to several equilibrium paths, some of which are not expected for linear elastic materials or found in the existing literature on nonlinear materials, resulting in several coexisting stable and unstable solutions and a complex potential energy landscape, thus clarifying the influence of the constitutive hyperelastic model on the results. Analytical expressions for the normalized snap-through and pitchfork bifurcation loads are derived as a function of the material parameters, truss geometry and imperfections for practical applications. The influence of Eulerian buckling on the truss load carrying capacity is also investigated and formulas to evaluate the buckling load under both vertical and horizontal loads are derived. The present results may help in the development of new engineering applications where multistability and large deformations are desired.

... From the point of view of the deformation mechanism occurring at the interaction of the auxetic material and mechanical energy [9], auxetic metamaterials are usually classified as re-entrant [10,11], chiral [12,13], rotating [14,15] and laminated with a hierarchical structure [16,17]. Such cellular structures are designed using complex algorithms and computer procedures [10,18,19] and manufactured using a 3D printing technique. ...

Auxetic structures exhibit unusual changes in size, expanding laterally upon stretching instead of contracting. This paper presents this effect in a failsafe mode in structures made of rigid squares. We applied the concept of auxetic structures made of rigid rotating squares (from Grima and Evans) and offer a novel solution for connecting them. By introducing axes of rotation on the surface of the squares, a reliable working system is obtained, free from stress, in which the squares can come into contact with each other and completely cover the surface of the structure, or, in the open position, form regularly arranged pores. Herein, we present a new 2D auxetic metamaterial that is mathematically generated based on a theoretical relationship of the angle between the edges of a square and the position of the axis of rotation. Physical models were generated in the form of a planar structure and in the form of a circular closed structure. Such physical models confirmed our initial considerations and the geometrical relationships, offering new application possibilities. The novel structure that was designed and manufactured for the purpose of the paper can be considered as a new proposal in the market of auxetic materials.

... Since then, diverse studies were dedicated to understanding the influences of the loading scenarios, material nonlinearities, pore arrangements and geometries, porosities, pattern transitions, etc. on the pattern formation and the accompanied bandgaps in soft metamaterials. Overvelde et al. [24] discussed the role of pore shape in controlling the compaction performance through buckling based on both experimental and numerical results and clarified that the circular hole is not an optimal shape. Wang et al. [25] revealed the effects of geometrical and material nonlinearities, as well as the applied strain on the evolution of bandgaps by considering an equi-biaxial compression and by employing the neo-Hookean and Gent material constitutions. ...

This paper proposes a new metamaterial structure consisting of periodically porous elastomers with pore coatings. This design enables us to engender finite deformation by a contactless load. As a case study, we apply thermal load to the surface coating and carry out a finite element analysis to probe instabilities and the associated phononic properties. It turns out that a novel buckling mode, preserving the nature of surface wrinkling in tubular structures, can be induced under a plane-strain setup, and a smaller size of the unit cell is attained compared to the counterpart of traditional buckled profile in soft porous elastomers. In particular, this buckling pattern is able to produce several bandgaps in different frequency ranges as the macroscopic mean strain increases. We further introduce a metallic core as a local resonator, and the updated metamaterial allows a low-frequency bandgap, the bandgap width of which can be estimated by a simplified theoretical model. As more free parameters are involved in the structure, we perform a detailed parametric study to elucidate the influences of the modulus ratio between coating and matrix, the porosity, the core radius, and the macroscopic mean strain on the buckling initiation and the evolution of bandgap. Remarkably, a stiffer surface coating is prone to enhance the stability of the structure, which is contrary to existing results in film/substrate bilayers. It is expected that the current study could shed light on new insight into pattern formation and wave manipulation in porous elastomers.

... Within the context of mechanical design, this trend has led to the emergence of "machine-matter", where architected materials exhibit some or most of properties and functionalities that have been traditionally imputed to machines. On the one side of the spectrum, mechanical metamaterials [1][2][3] are rationally designed to exhibit exotic material-like properties, such as negative effective values of the Poisson's ratio [4][5][6] , thermal expansion [7][8][9] , and stiffness 10,11 . On the other side of the spectrum, however, one finds such concepts as mechanical logic gates 12,13 , adaptive-stiffness mechanisms 14,15 , and shape-shifting designs [16][17][18][19] , which exhibit device-like functionalities. ...

Machine-matter, of which mechanical metamaterials and meta-devices are important sub-categories, is emerging as a major paradigm for designing advanced functional materials. Various exciting applications of these concepts have been recently demonstrated, ranging from exotic mechanical properties to device-like and adaptive functionalities. The vast majority of the studies published to date have, however, focused on the quasi-static behavior of such devices, neglecting their rich dynamic behavior. Recently, we proposed a new class of strain rate-dependent mechanical metamaterials that are made from bi-beams (i.e., viscoelastic bilayer beams). The buckling direction of such bi-beams can be controlled with the applied strain rate. The proposed approach, however, suffers from a major limitation: 3D printing of such bi-beams with such a 'strong' differential strain rate-dependent response is very challenging. Here, we propose an alternative approach that only requires a 'weak' differential response and a rationally designed geometric artifact to control the buckling direction of bi-beams. We present an analytical model that describes the landscape of all possible combinations of geometric designs and hyperelastic as well as viscoelastic properties that lead to the desired strain rate-dependent switching of the buckling direction. We also demonstrate how multi- and single-material 3D printing techniques can be used to fabricate the proposed bi-beams with microscale and submicron resolutions. More importantly, we show how the requirement for a weak differential response eliminates the need for multi-material 3D printing, as the change in the laser processing parameters is sufficient to achieve effective differential responses. Finally, we use the same 3D printing techniques to produce strain rate-dependent gripper mechanisms as showcases of potential applications.

... Some of them that are reviewed here are bucklicrystal structure, sinusoidal filament networks, and cross-chiral structures. In 2012, Overvelde et al. 143 demonstrated buckli structures of 2D soft materials with different void shapes like circle, square, triangular, trihexagonal, and rhombi-trihexagonal tessellations ( Figure 7a). The auxetic behavior obtained upon submitting these materials to external uniaxial compression originated from their buckled structure, resulting in a minimum NPR value of −0.39 and a maximum NPR value of −0.95 for triangular and square shape voids, respectively. ...

Over the last three decades but more particularly during the last 5 years, auxetic mechanical metamaterials constructed from precisely architected polymer-based materials have attracted considerable attention due to their fascinating mechanical properties. These materials present a negative Poisson's ratio and therefore unusual mechanical behavior, which has resulted in enhanced static modulus, energy adsorption, and shear resistance, as compared with the bulk properties of polymers. Novel advanced polymer processing and fabrication techniques, and in particular additive manufacturing, allow one to design complex and customizable polymer architectures that are particularly relevant to fabricate auxetic mechanical metamaterials. Although these metamaterials exhibit exotic mechanical properties with potential applications in several engineering fields, biomedical applications seem to be one of the most relevant with a growing number of articles published over recent years. As a result, special focus is needed to understand the potential of these structures and foster theoretical and experimental investigations on the potential benefits of the unusual mechanical properties of these materials on the way to high performance biomedical applications. The present Review provides up to date information on the recent progress of polymer-based auxetic mechanical metamaterials mainly fabricated using additive manufacturing methods with a special focus toward biomedical applications including tissue engineering as well as medical devices including stents and sensors.

... The buckling of lattice structures depends on multi-parameters, such as wall thickness, tessellation, relative density, presence of column inside the structure, lattice pore size, and length of the column [37,[46][47][48][49]. Figure 8 represents the local and global buckling behavior of all uniaxial compressive tested columns. For each type of sample, only one specimen is presented. ...

Lattice structures possess many superior properties over solid materials and conventional structures. Application-oriented lattice structure designs have become a choice in many industries, such as aerospace, automotive applications, construction, biomedical applications, and footwear. However, numerical and empirical analyses are required to predict mechanical behavior under different boundary conditions. In this article, a novel surface-based structure named O-surface structure is designed and inspired by existing Triply Periodic Minimal Surface morphologies in a particular sea urchin structure. For comparison, both structures were designed with two different height configurations and investigated for mechanical performance in terms of compression, local buckling, global buckling, and post-buckling behavior. Both simulation and experimental methods were carried out to reveal these aforementioned properties of samples fabricated by multi jet fusion technology. The sea urchin structure exhibited better mechanical strength than its counterpart, with the same relative density almost two-folds higher in the compressive response. However, the O-surface structure recorded more excellent energy absorption and flexible behavior under compression. Additionally, the compression behavior of the O-surface structure was progressive from top to bottom. In contrast, the sea urchin structure was collapsed randomly due to originated cracks from unit cells’ centers with local buckling effects. Moreover, the buckling direction of structures in long columns was also affected by keeping the relative density constant. Finally, based on specific strength, the O-surface structure exhibited 16-folds higher specific strength than the sea urchin structure.

... The contraction under compression leads to an increase in density, which makes them suitable for structures with high indentation resistance, 12 shear resistance, 13 energy absorption, 14 fracture toughness, 15 drug delivery, and wound management. 16 Different designs possessing auxetic behavior have been introduced, e.g., buckling of porous structures, 17 bucklicrystals, 18 and re-entrant, or bowtie, structures. 19,20 Another class of materials with negative Poisson's ratio are chiral-based structures. ...

Mechanical metamaterials with zero or negative Poisson’s ratio were subject to increasing research interest over the last few years. Their energy absorption capabilities make them suitable for impact and dampening applications, such as personal protection equipment or packaging materials. The variable porosity and unusual mechanical properties also make them applicable in drug delivery systems and wound management. Herein, we present an extension to common auxetic structures, including tetra-chirals and tetra-antichirals. By introducing an asymmetry in the design of their unit cell, Poisson’s ratio can be varied over a broad range. Specimens with a selected amount of asymmetry were additively manufactured with a thermoplastic polyurethane using fused filament fabrication. Compression tests were performed to investigate the influence of the asymmetry on Poisson’s ratio and the compression modulus. Two different numerical models were employed using ABAQUS to describe the mechanical properties of the structures and were verified by the experiments. The numerical models are based on three-point bending test data. Both asymmetric designs show an influence of the asymmetry onto Poisson’s ratio, resulting in variable Poisson’s ratio, porosity, and compression modulus.

Recent attention to pneumatically pressurized mechanical metamaterials has identified opportunity for large shape change and mechanical properties adaptation through the collective exploitation of reconfigurable internal structures and enclosed cavities. Yet, many of these ideas are found to act in smooth, continuous ways at moderate rate. This research explores a new class of bimodal, hierarchical mechanical metamaterials that exemplify rapid change of mechanical behavior by exploiting pneumatic pressure as a means to cross bifurcation. A lattice structure with periodic square cavities is presented as a model metamaterial to highlight important design considerations and mechanical behavior. An analytical model is formed to delineate the multimodal boundaries in a high dimensional parameter space, while numerical simulations and experiments confirm the presence of each modal characteristic. The studies reveal the high rate of change afforded by this embodiment of pressurized mechanical metamaterials and confirm the origin lay in harnessing elastic instability. Extensions to the idea are compared against the Kutzbah-Grübler criteria to articulate how other metamaterial networks may be leveraged in this way. The outcomes of this research may inspire methods for high rate shape and properties change via multimodal mechanical metamaterial assemblies, such as for soft robotic platforms.
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Phononic crystals (PnCs) attract attentions in acoustic energy conversion and harvesting due to their excellent properties in regulating elastic and acoustic waves. Topological imbalance and semi-enclosed defect has been proposed in this work to systematically broaden the band gap of PnCs and improve the localization of elastic wave in defect. Parametric equations are introduced to generate nine square lattice unit cells with four-fold rotationally symmetric shapes. The imbalance between the long and short branches is utilized to describe the topological evolution of the perfect PnCs. With the increase of topological imbalance, the jump of the Bloch mode appears and thus leads to the band gap opening and expansion. The defect supercell with the highest topological imbalance has the best energy confinement effect. The 2D and 3D transient numerical simulations for the PnC of the highest topological imbalance indicate that the energy confinement effect of the semi-enclosed line defect is more intensive than the semi-enclosed arrow or bottle defect, which is confirmed by piezoelectric energy harvesting experiments. Under the excitation of 50 kHz, the period-2 nonlinear phenomena of the output voltage by piezoelectric disk are experimentally noticed in the PnC with the semi-enclosed line defect. The peak-to-peak power is 3.08 mW at the optimal resistance. Compared with the traditional point defect case, the voltage of the semi-enclosed line defect case is increased by 12.0 times and its power is increased by 75.1 times due to combination of perfect mirror effect and nonlinear defect state mechanism. This study provides a new avenue in design of high-frequency nonlinear acoustic devices and self-powered acoustic sensors.

Buckling is one type of unstable structural response which has been known for centuries. Although buckling is generally regarded as a type of undesirable phenomenon, many investigations reported that it could be utilized to generate buckling‐induced metamaterials and structures with negative Poisson's ratio. Recent studies demonstrated that buckling‐induced auxetic behavior would disappear when the base material was changed from elastomer to metal. A pattern scale factor (PSF) method to recover auxeticity of metallic metamaterials and structures was mentioned in the previous studies, but the method was not specifically introduced. Here, a more detailed introduction of the PSF method is made and some successful case studies are presented. This article is protected by copyright. All rights reserved.

Magnetic soft materials (MSMs) have shown potential in soft robotics, actuators, metamaterials, and biomedical devices because they are capable of untethered, fast, and reversible shape reconfigurations as well as controllable dynamic motions under applied magnetic fields. Recently, magnetic shape memory polymers (M-SMPs) that incorporate hard magnetic particles in shape memory polymers demonstrated superior shape manipulation performance by realizing reprogrammable, untethered, fast, and reversible shape transformation and shape locking in one material system. In this work, we develop a multimaterial printing technology for the complex structural integration of MSMs and M-SMPs to explore their enhanced multimodal shape transformation and tunable properties. By cooperative thermal and magnetic actuation, we demonstrate multiple deformation modes with distinct shape configurations, which further enable active metamaterials with tunable physical properties such as sign-change Poisson's ratio. Because of the multiphysics response of the M-MSP/MSM metamaterials, one distinct feature is their capability of shifting between various global mechanical behaviors such as expansion, contraction, shear, and bending. We anticipate that the multimaterial printing technique opens new avenues for the fabrication of multifunctional magnetic materials.

Stress concentration in porous materials is one of the most crucial culprits of mechanical failure. This paper focuses on planar porous materials with porosity less than 5%. We present a stress-prediction model of an arbitrarily rotated elliptical hole in a rhombus shaped representative volume element (RVE) that can represent a class of generic planar tessellations, including rectangular, triangular, hexagonal, Kagome, and other patterns. The theoretical model allows the determination of peak stress and distribution of stress generated near the edge of elliptical holes for any arbitrary tiling under displacement loading and periodic boundary conditions. The results show that the alignment of the void with the principal directions minimizes stress concentration. Numerical simulations support the theoretical findings and suggest the observations remain valid for porosity as large as 5%. This work provides a fundamental understanding of stress concentration in low-porosity planar materials with insight that not only complements classical theories on the subject but also provides a practical reference for material design in mechanical, aerospace, and other industry.

From designing architected materials to connecting mechanical behavior across scales, computational modeling is a critical tool for understanding and predicting the mechanical response of deformable bodies. In particular, computational modeling is an invaluable tool for predicting global emergent phenomena, such as the onset of geometric instabilities, or heterogeneity induced symmetry breaking. Recently, there has been a growing interest in both using machine learning based computational models to learn mechanical behavior directly from experimental data, and using machine learning (ML) methods to reduce the computational cost of physics-based simulations. Notably, machine learning approaches that rely on Graph Neural Networks (GNNs) have recently been shown to effectively predict mechanical behavior in multiple examples of particle-based and mesh-based simulations. However, despite this initial promise, the performance of graph based methods have yet to be investigated on a myriad of solid mechanics problems. In this work, we examine the ability of neural message passing to predict a fundamental aspect of mechanically driven emergent behavior: the connection between a column’s geometric structure and the direction that it buckles. To accomplish this, we introduce the Asymmetric Buckling Columns (ABC) dataset, a dataset comprised of three types of asymmetric and heterogeneous column geometries (sub-dataset 1, sub-dataset 2, and sub-dataset 3) where the goal is to classify the direction of symmetry breaking (left or right) under compression after the onset of the buckling instability. Notably, it is difficult to parameterize these structures into a feature vector for typical ML methods. Essentially, because the geometry of these columns is discontinuous and intricate, local geometric patterns will be distorted by the low-resolution “image-like” data representations that are required to implement convolutional neural network based metamodels. Instead, we present a pipeline to learn global emergent properties while enforcing locality with message passing neural networks. Specifically, we take inspiration from point cloud based classification problems from the computer vision research field and use PointNet++ layers to perform classification on the ABC dataset. In addition to investigating GNN model architecture, we study the effect of different input data representation approaches, data augmentation, and combining multiple models as an ensemble. Overall, we were able to achieve good performance with this approach, ranging from 0.952 prediction accuracy on sub-dataset 1, to 0.913 prediction accuracy on sub-dataset 2, to 0.856 prediction accuracy on sub-dataset 3 for training dataset sizes of 20,000 points each. However, these results also clearly indicate that predicting solid mechanics based emergent behavior with these methods is non-trivial. Because both our model implementation and dataset are distributed under open-source licenses, we hope that future researchers can build on our work to create enhanced mechanics-specific machine learning methods. Furthermore, we also intend to provoke discussion around different methods for representing complex mechanical structures when applying machine learning to mechanics research.

2D soft periodic structures can produce reversible compaction and relaxation behaviors with complex and diverse compacting patterns during loading and unloading. The (trans)formation and control of the recoverable and adjustable compacting pattern are essential in their wide range of application prospects. In addition to the traditional schemes, this paper proposes and discusses the possibility of using combined loading to control the compacting pattern of finite-size 2D soft periodic structures. Based on the detailed analysis and design of the loading schemes, the buckling and post-buckling behaviors of four kinds of 2D soft periodic structures with finite size under biaxial loading and combined normal and shear loading are investigated by a validated ABAQUS finite element analysis. The buckling model shapes of each structure under each loading combination are discussed and classified in detail to determine the Representative Compacting Patterns (RCPs) and the Critical Points of model transformation Controlled by Loading (CPCL) on the loading plane. Of course, some new modes have been found. Moreover, the CPCL between RCPs without conversion relationship is found to have a corresponding relationship with the variation of the difference between the first eigenvalue and the second eigenvalue of the structure on the loading plane. By using the rotation angle of the hole walls as an index, the compacting behavior and characteristics of each pattern in the process of post-buckling are analyzed quantitatively. The research ideas and results of this paper will provide a new design space for various application fields of finite-size 2D soft periodic structures.

Auxetic metamaterials with negative Poisson’s ratio have attracted much attention due to its counterintuitive deformation behavior over the conventional engineering materials. However, it is difficult to describe the complex correlation between microstructure parameters and auxeticity by analytical or empirical solutions in the form of math expressions. In this work, the machine learning (ML) model with artificial neural network (ANN) is developed to analyze a novel planar auxetic metamaterial designed by introducing orthogonally aligned oval‐shaped perforations in solid base material and its feasibility is demonstrated through the experimental and FEM solutions. It is found that the proposed structure involving less design parameters exhibits the best performance at the aspects of auxetic behavior and stress level than those with peanut‐shaped holes and elliptic holes. Moreover, the results of parameter analysis demonstrate that the present ML solution model can provide accurate predicting results rapidly for this problem, without the limitations of explicit solution expressions which are typically not available in practice. The ML model allows one to obtain the desired auxetic property by tailoring the geometric parameters effectively and accelerate auxetic metamaterial design.
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This study presents the compression response of additively manufactured novel soft porous structures with architected microstructure. Six porous additively manufactured architected periodic structures with two‐fold and four‐fold symmetry were considered. The effect of pore shape and fold symmetry of microstructure on the non‐linear response of a square array of architected pores in a soft polymeric matrix is experimentally investigated. The digital image correlation (DIC) is used for investigating the evolution of strains and deformation during uniaxial tensile tests and compression tests of porous structures. Compression induced instability lead to negative Poisson's ratio, and compaction of porous structures, which is found to depend not only on the shape of the architecture but also the fold symmetry exists in the microstructure's unit cell. Unique architectures with multiple buckling modes and shape transformation are also observed. Two‐fold symmetry structures are found to buckle at lower strains compared to the four‐fold symmetric structure at the same porosity level and produced high compaction and negative Poisson's ratio. The results showed that in addition to pore shape, the fold symmetry could be used effectively to design a new class of soft, active, and reconfigurable devices over a wide range of length scales with desired characteristics. This study investigates the compressibility and auxeticity of a new family of soft porous architected periodic structures with two‐fold and four‐fold symmetries. The results show that in addition to pore shape, the two‐fold symmetry can be used effectively to design soft reconfigurable devices over a wide range of length scales with higher compressibility and negative Poisson's ratio.

Architected elastomeric beam networks have great potential for energy absorption, multi-resonant vibration isolation, and multi-bandgap elastic wave control, due to the reconfigurability and programmability of their mechanical buckling instabilities. However, navigating this design space is challenging due to bifurcations between mono- and bistable beam designs, inherent geometric nonlinearities, and the strong dependence of buckling properties on beam geometry. To investigate these challenges, we developed a Bayesian optimization framework to control the equilibrium states of an inclined elastomeric beam, while also tuning the energy to transition between these configurations. Leveraging symmetry to reduce the design space, the beam shape is parameterized using a Fourier series representation. A penalty method is developed to include monostable designs in objective functions with dependencies on bistable features, enabling monostable results to still be incorporated in the Gaussian Process surrogate and contribute to the optimization process. Two objectives are optimized in this study, including the position of the second stable equilibrium configuration and the ratio of output to input energy between the two stable states. A scalarized multi-objective optimization is also carried out to study the trade-off between equilibrium position and the energetics of transition between the stable states. The predicted designs are qualitatively verified through experimental testing. Collectively, the study explores a new parameter space for beam buckling, introduces a penalty method to regularize between mono- and bistable domains and provides a library of beams as building blocks to assemble and analyze in future studies.

Mechanical metamaterials can be defined as a class of architected materials that exhibit unprecedented mechanical properties derived from designed artificial architectures rather than their constituent materials. While macroscale and simple layouts can be realized by conventional top-down manufacturing approaches, many of the sophisticated designs at various length scales remain elusive, due to the lack of adequate manufacturing methods. Recent progress in additive manufacturing (AM) has led to the realization of a myriad of novel metamaterial concepts. AM methods capable of fabricating microscale architectures with high resolution, arbitrary complexity, and high feature fidelity have enabled the rapid development of architected metamaterials and drastically reduced the design-computation and experimental-validation cycle. This paper first provides a detailed review of various topologies based on the desired mechanical properties, including stiff, strong, and auxetic (negative Poisson’s ratio) metamaterials, followed by a discussion of the AM technologies capable of fabricating these metamaterials. Finally, we discuss current challenges and recommend future directions for AM and mechanical metamaterials.

The Poynting effect generically manifests itself as the extension of the material in the direction perpendicular to an applied shear deformation (torsion) and is a material parameter hard to design. Unlike isotropic solids, in designed structures, peculiar couplings between shear and normal deformations can be achieved and exploited for practical applications. Here, a metamaterial is engineered that can be programmed to contract or extend under torsion and undergo nonlinear twist under compression. First, it is shown that the system exhibits a novel type of inverted Poynting effect, where axial compression induces a nonlinear torsion. Then the Poynting modulus of the structure is programmed from initial negative values to zero and positive values via a pre‐compression applied prior to torsion. The work opens avenues for programming nonlinear elastic moduli of materials and tuning the couplings between shear and normal responses by rational design. Obtaining inverted and programmable Poynting effects in metamaterials inspires diverse applications from designing machine materials, soft robots, and actuators to engineering biological tissues, implants, and prosthetic devices functioning under compression and torsion. A rationally designed meta‐cylinder exhibits inverted and programmable Poynting effects. It shows nonlinear torsions under compression (inverted Poynting) and contractions under torsion. By applying a level of pre‐compression the structure can be programmed to induce tunable contraction or dilation (negative or positive Poynting effect) when twisted.

Despite a great variety of underlying mechanisms, the overall behavior of multistable systems is fairly similar. All of them exhibit very complex dynamics due to nonlinear interactions leading to the coexistence of attractors. A particular feature of such systems is their extremely high sensitivity to initial conditions. Even a slight change in the initial condition can lead the system to a different attractive state. This sensitivity is especially pronounced in multistable systems with interwoven basins of attraction. Furthermore, multistable systems are very sensitive to parameter perturbations that can cause a qualitatively different behavior, especially near the bifurcation point, where a tiny change in a control parameter may result in the emergence of a large number of attractors.

In this article, we report that, machine learning, an artificial intelligent technique, is used to optimize biomimetic rods and lattice structures. Various structures available in nature such as plant stems and roots that exhibit better buckling resistance are mimicked and modeled using finite element analysis, to obtain a training dataset. For validating the finite element analysis, uniaxial compression to buckling of additive manufactured biomimetic rods using a polymeric ink is performed. These model results are then formed into a dataset. Forward design and data filtering are conducted by machine learning to optimize the biomimetic rods from the dataset. The results show that the machine learning assisted rod designs have 150% better buckling resistance than all the rods in the training dataset, i.e., better than the nature’s counterparts. These optimal rods can be used in designing structures with superior buckling resistance such as in bridges, buildings, lattice structures, etc. Using these biomimetic rods, lattice structures with better structural performance are manufactured. While lattice unit cells such as octahedron, tetrahedron, octet, etc., have been previously proposed for lightweight structures, it is plausible that more optimal unit cells exist which might perform better than the existing counterparts. Machine learning technique is used to discover new optimal cells. Uniaxial compression tests using ANSYS are performed to form a dataset, which is used to train machine learning algorithms and form predictive model. The predictive model is then used to identify a total of 20 optimal symmetric unit cells. These new unit cells show 51%–57% higher capacity than octet cell. Particularly, if the porous biomimetic rods are used to construct the unit cells, an additional 130%–160% increase in buckling resistance is achieved. New lattice unit cells exhibit a buckling load of 261%–308% higher than the classical octet unit cell. Sandwich structures manufactured by 3D printing these optimal symmetric unit cells show 13%–35% higher flexural strength. This study opens up new opportunities to design high-performance metamaterials combining biomimetics and machine learning.

Soft cellular mechanical metamaterials (CMMs) have gained increasing attention due to their unique mechanical properties, especially when under large deformation. However, the strong nonlinearities and complex instabilities brought by the large deformation field is a critical challenge for the rational design of soft CMMs. The rational design of soft CMMs are challenging though, due to strong nonlinearity and instabilities often stemmed from large deformation conditions. especially under large deformation, where often take place. In this work, we propose a mapped shape optimization method as a computational framework for inverse designs of soft CMMs. The core of this method is to introduce a fixed referential configuration. The geometric changes of the cellular structures are reflected by altering a differentiable shape map; and the deformation of the corresponding structures are determined by mapping the finite element computations to the referential configuration. Such formulation avoids the need to alter the background mesh and more importantly, provides an efficient way to compute the gradient of the objective functions with respect to the design variables via the adjoint method. The proposed method is of general purpose, and three distinct yet representative numerical examples are used to demonstrate the effectiveness of the method: optimizing unique overall mechanical properties, precise control of the onset of instability, and optimizing phononic band gaps. These examples cover a broad range of important engineering applications of soft CMMs.

The 3D auxetic buckling pattern in the compression process is used to design auxetic structures. However, obtaining this pattern is time consuming and unintentional. In this paper, three auxetic structures, namely, globally deformed cross-frustum (GC) structure, partially deformed cross-frustum (PC) structure, and core deformed cross-frustum (CC) structure are proposed on the basis of the 3D auxetic deformation equation. The GC structure and PC structure can stably realize the 3D auxetic property through finite element analysis and experimental verification of structures. The structural strength of the latter structure is 2–5 times greater than that of the former structure. The CC structure exhibits a 3D auxetic property during the entire compression process and has excellent structural strength, which is greater than that of the first two structures. The rotational pattern can be also realized by the CC structure within the specified parameter range.
A soft crawling robot with a motion module and perception module is designed by using the proposed auxetic structures. The motion module of the soft robot is designed by using the GC structure, and the perception module is realized by using the CC structure. The combination of modules can realize different functions, including advancement and rotation. The robot can be rapidly fabricated by digital light processing and has potential application value in medical treatment and other fields.

Metamaterial has received great interest during the last ten years in distinct field, owing of its key characteristics such as enhancement in bandwidth, radiated power, directivity and controls the direction of electromagnetic radiation. It is a smart or a new class of manmade invented materials that can achieve electromagnetic properties that do not occur naturally, such as electromagnetic cloaking or negative index of refraction. These materials exhibit negative value of permittivity, permeability and refractive index. In addition, metamaterial extract their properties from their structure rather than the material of which they are composed of. Further, these materials exhibit excellent design flexibility with their customized properties and their tunability under external stimuli. Due to these attractive properties, metamaterials have enabled the development of new devices and concepts and possible utilization in diverse novel applications. Thus, metamaterials find wide application in medical sector, automotive, aerospace, and many other devices (biosensor, crowd control, absorbers, antennas, optical filters, infrastructure monitoring, smart solar power management, energy harvesters and even shielding structure from earthquakes etc). Hence the aim of this paper is to provide a comprehensive overview of different metamaterial, historical development and their applications in different sectors. In addition, perspective about the challenges and future scope for development of metamaterial is also presented, so that this article could become the torch bearer for the new researchers working in the area of advanced materials.

A mathematical model of multiple layer skin coloration in cephalopods, a class of aquatic animals, is presented. The model incorporates diffuse and specular reflection from both pigment and structural photonic components found in the skin of these animals. Specific physical processes of this coloration are identified and modeled utilizing available biological materials data. Several examples of combination spectra are calculated to illustrate multiple layer and incident light effects as well as the potentially rich repertoire of color schemes available to these animals. A detailed understanding of the physical principles underlying cephalopod coloration is expected to yield insights into their possible functions.

The rapid closure of the Venus flytrap (Dionaea muscipula) leaf in about 100 ms is one of the fastest movements in the plant kingdom. This led Darwin to describe the plant as "one of the most wonderful in the world". The trap closure is initiated by the mechanical stimulation of trigger hairs. Previous studies have focused on the biochemical response of the trigger hairs to stimuli and quantified the propagation of action potentials in the leaves. Here we complement these studies by considering the post-stimulation mechanical aspects of Venus flytrap closure. Using high-speed video imaging, non-invasive microscopy techniques and a simple theoretical model, we show that the fast closure of the trap results from a snap-buckling instability, the onset of which is controlled actively by the plant. Our study identifies an ingenious solution to scaling up movements in non-muscular engines and provides a general framework for understanding nastic motion in plants.

Periodic elastomeric cellular solids are subjected to uniaxial compression and novel transformations of the patterned structures are found upon reaching a critical value of applied load. The results of a numerical investigation reveal that the pattern switch is triggered by a reversible elastic instability. Excellent quantitative agreement between numerical and experimental results is found and the transformations are found to be remarkably uniform across the samples. It is proposed that the mechanism will also operate at much smaller scales opening the possibility for imprinting complex patterns at the nanoscale or switching photonic and phononic crystals in a controlled way.

Although buckling instabilities in elastic solids have been known for a long time, high interest in this phenomenon is relatively recent. The current and prospective applications in flexible electronics, materials with tunable surface properties (adhesion and wettability), responsive photonic and phononic structures, and reinforced nanocomposites led to a surge in the interest in buckling instabilities. In fact, some of the applications, such as flexible electronics and metrology, have advanced at a tremendous pace only within the past few years. In this review, we discuss some of the most recent progress in the fundamental understanding of buckling instabilities in periodic multi-component polymer materials and porous polymer structures. We also discuss how the buckling can be localized to predetermined regions and hence form periodic instability patterns. Finally, we present several recent examples where buckling instabilities have been employed as a patterning tool to realize complex surface arrays of various materials.

Cellular solids include engineering honeycombs and foams (which can now be made from polymers, metals, ceramics, and composites) as well as natural materials, such as wood, cork, and cancellous bone. This new edition of a classic work details current understanding of the structure and mechanical behavior of cellular materials, and the ways in which they can be exploited in engineering design. Gibson and Ashby have brought the book completely up to date, including new work on processing of metallic and ceramic foams and on the mechanical, electrical and acoustic properties of cellular solids. Data for commercially available foams are presented on material property charts; two new case studies show how the charts are used for selection of foams in engineering design. Over 150 references appearing in the literature since the publication of the first edition are cited. It will be of interest to graduate students and researchers in materials science and engineering. © Lorna J. Gibson and Michael F. Ashby, 1988 and Lorna J. Gibson and Michael F. Ashby, 1997.

It is of great importance for the development of new products to find the best possible topology or layout for given design objectives and constraints at a very early stage of the design process (the conceptual and project definition phase). Thus, over the last decade, substantial efforts of fundamental research have been devoted to the development of efficient and reliable procedures for solution of such problems. During this period, the researchers have been mainly occupied with two different kinds of topology design processes; the Material or Microstructure Technique and the Geometrical or Macrostructure Technique. It is the objective of this review paper to present an overview of the developments within these two types of techniques with special emphasis on optimum topology and layout design of linearly elastic 2D and 3D continuum structures. Starting from the mathematical-physical concepts of topology and layout optimization, several methods are presented and the applicability is illustrated by a number of examples. New areas of application of topology optimization are discussed at the end of the article. This review article includes 425 references.

Some engineering applications require structures to expand and contract in size, while retaining their exterior shape. The applications range from mundane daily life objects to more fancy art structures. In contrast to a multi degree-of-freedom structure, a single degree-of-freedom structure can be driven by a single actuator reducing cost and simplifying the control. In this paper we study single degree-of-freedom structures that can be formed by a lattice of single degree-of-freedom polyhedral expanding units. Due to built-in symmetries, the entire structure can expand and contract as one of the units in the structure is actuated. The paper describes the design of polyhedral single degree-of-freedom systems, the structures of their dynamics/optimal control, and results from construction prototypes.

The photonic band structure and optical transmittance of two-dimensional periodic elastomeric photonic crystals are studied computationally to understand the effects of large strains on optical properties of the structures. The large compressive deformation patterns of the two-dimensional periodic structure studied by Mullin and coworkers [Mullin, T., Deschanel, S., Bertoldi, K., Boyce, M.C., 2007. Pattern transformation triggered by deformation. Physical Review Letters 99(8), 084301] are first reproduced using hyperelastic material models for the elastomer SU-8. Finite element analysis is then used to solve Maxwell's equations to obtain light transmittance through both the undeformed and deformed structures; simultaneously the wave equation resulting from the appropriate two-dimensional form of Maxwell's equations is solved as an eigenvalue problem to obtain the band structure. The deformation-induced shift in transmission spectrum valleys for different bands is calculated, and the changes in the width of these reflectance peaks are also obtained. The band structure calculation shows that there are no complete photonic band gaps as expected for the low dielectric contrast system. However, the effect of the observed reversible, symmetry-breaking deformation pattern is to uncouple many of the photonic bands in all three high symmetry directions, i.e. Γ–X, X–M, and Γ–M. New non-degenerate deformation-induced optical modes appear in both the real space transmittance spectra and the band structure with lower reflectance values. Analyses of the deformation pattern, the optical mode shapes, and the photonic band structure reveal that localized regions of large rotation are responsible for the significant changes in optical transmittance. The results have practical importance for the design of strain-tunable optomechanical materials for sensing and actuation.

A class of expandable polyhedral structures, the expandohedra, consisting of prismatic faces linked along edges by hinged plates, provides a model for the swelling of viruses. The finite breathing mode of the expandohedra, apparent in the model, and from geometric arguments, is not, however, detected by elementary counting techniques. Symmetry extended mobility criteria show that the breathing motion is part of a set of face mechanisms for many expandohedra. Numerical analysis using the singular value decomposition of the compatibility matrix confirms the completeness of the deductions from general symmetry theorems, derived for expandohedra.

This paper is a survey of structural shape optimization with an emphasis on techniques dealing with shape optimization of the boundaries of two- and three-dimensional bodies. Attention is focused on the special problems of structural shape optimization which are due to a finite element model which must change during the optimization process. These problems include the requirement for sophisticated automated mesh generation techniques and careful choice of design variables. They also include special problems in obtaining sufficiently accurate sensitivity derivatives.

Many structural and functional properties possessed by plants have great potentials to stimulate new concepts and innovative ideas in the field of biomimetic engineering. The key inputs from biology can be used for creation of efficient and optimized structures. The study of the geometry and folding pattern of leaves of Mimosa pudica, referred as Sensitive Plant, reveals some of the peculiar characteristics during folding and unfolding. When the leaf is touched, it quickly folds its leaflets and pinnae and droops downward at the petiole attachment. With the help of experiments on simulation model, the variations in angle of leaflets and degree of compaction after folding are investigated.

Pattern transformation in periodic microporous elastoplastic solid coatings is caused by a buckling of the struts and a rotation of the nodes under compressive stresses. The results of a nonlinear numerical investigation confirm the critical role of the bifurcation of the periodic solid under compressive stresses. In striking contrast to the earlier observations of elastic instabilities in porous elastomeric solids, the elastic-plastic nature of the cross-linked periodic microstructure studied here provides the ability to lock in the transformed pattern with complete relaxation of the internal stresses. The study unveils a novel deformation mode in porous periodic solids in the form of organized buckling instability of weak strut elements.

Negative Poisson's ratio behavior has been uncovered in cellular solids that comprise a solid matrix with a square array of circular voids. The simplicity of the fabrication implies robust behavior which is relevant over a range of scales. The behavior results from an elastic instability which induces a pattern transformation and excellent quantitative agreement is found between calculation and experiment.

We report a fully reversible and robust shape-memory effect in a two-dimensional nanoscale periodic structure composed of three steps, the elastic instability governing the transformation, the plasticity that locks in the transformed pattern as a result of an increase in glass transition temperature (T(g)), and the subsequent elastic recovery due to the vapor-induced decrease in T(g). Solvent swelling of a cross-linked epoxy/air cylinder structure induces an elastic instability that causes a reversible change in the shape of the void regions from circular to oval. The pattern symmetry changes from symmorphic p6mm to nonsymmorphic p2gg brought via the introduction of new glide symmetry elements and leads to a significant change in the phononic band structure, specifically in the opening of a new narrow-band gap due to anticrossing of bands, quite distinct from gaps originating from typical Bragg scattering. We also demonstrate that numerical simulations correctly capture the three steps of the shape-memory cycle observed experimentally.

We show how to employ an interference lithographic template (ILT) as a facile mold for fabricating three-dimensional bicontinuous PDMS (poly(dimethylsiloxane)) elastomeric structures and demonstrate the use of such a structure as a mechanically tunable PDMS/air phononic crystal. A positive photoresist was used to make the ILT, and after infiltration with PDMS, the resist was removed in a water-based basic solution which avoided PDMS swelling or pattern collapse occurring during the ILT removal process. Since the period of the structure is approximately 1 microm, the density of states of gigahertz phonons are altered by the phononic PDMS/air crystal. Brillouin light scattering (BLS) was employed to measure phononic modes of the structure as a function of mechanical strain. The results demonstrate that the phononic band diagram of such structures can be tuned mechanically.

We report on a simple yet robust method to produce orientationally modulated two-dimensional patterns with sub-100 nm features over cm2 regions via a solvent-induced swelling instability of an elastomeric film with micrometer-scale perforations. The dramatic reduction of feature size ( approximately 10 times) is achieved in a single step, and the process is reversible and repeatable without the requirement of delicate surface preparation or chemistry. By suspending ferrous and other functional nanoparticles in the solvent, we have faithfully printed the emergent patterns onto flat and curved substrates. We model this elastic instability in terms of elastically interacting "dislocation dipoles" and find complete agreement between the theoretical ground-state and the observed pattern. Our understanding allows us to manipulate the structural details of the membrane to tailor the elastic distortions and generate a variety of nanostructures.

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