A review on mechanical metamaterials: Exploring negative Poisson's ratio,energy absorption,failure mechanisms,and current research frontiers

Abstract

Mechanical metamaterials have become increasingly attractive due to their numerous attributes, including low weight, high strength, negative Poisson's ratio, and high energy absorption. Traditionally, metamaterials have been utilised in wave manipulation such as in optics, acoustics and electromagnetism. Due to the mentioned attributes, there has been a notable rise in the adoption/research of these metamaterials for mechanical and structural applications. This article aims to present a review of the properties and state-of-the-art progress in mechanical metamaterials. A comprehensive review of the literature is presented, highlighting the state-of-the-art in negative Poisson's ratio (NPR), high energy absorption, failure mechanisms, design strategies, and 4D metamaterials. It is observed that the research is currently focusing on the creation/design of novel metamaterials and the enhancement of the properties of existing structures. Some of the common unit cells adopted for mechanical metamaterials include lattice-based, origami/Kirigami-inspired, cellular, or hierarchical unit cells. While great achievements can be reported on NPR and high-energy absorption metamaterials, there is a lag in the failure studies of mechanical metamaterials, especially on fracture toughness. Furthermore, there are concerted efforts to apply topology optimisation, numerical simulations, and machine learning to design higher-performing metamaterials. It is observed that machine learning and data-based design strategies are promising areas of research for the development of high-performance materials.

The review article is motivated by the absence of a comprehensive review on mechanical metamaterials despite many research outputs. Several gaps are presented, and this article will be a useful resource for the development of novel mechanical metamaterials.

Introduction

Historical background

Metamaterials, derived from the Greek word “meta”, meaning “beyond/superior,” are specially designed materials that exhibit properties which are not present in naturally occurring substances. ¹ They are generally composed of assemblies of repeating units created from materials such as metals, alloys, polymers, or composites. Metamaterials represent a paradigm shift in materials science, allowing scientists to design and develop materials with properties tailored for specific applications. The particular properties of metamaterials stem from their carefully designed, often periodic, microstructure rather than the intrinsic composition of the building materials. ² In other words, a metamaterial can be defined as an artificially designed material with unique properties arising from its structural arrangement, rather than the intrinsic characteristics of its constituents.

The development of metamaterials progressed significantly with the seminal work of John Pendry and David R. Smith (flowchart in Figure 1). ³ Building upon the theoretical foundations of the predictions of negative-refractive-index materials by Victor Veselago in 1967, Pendry proposed the theory of a “perfect lens” in 1999, demonstrating how metamaterials with a negative refractive index could overcome the diffraction limit in optics. This theoretical breakthrough has stimulated extensive research on the practical realisation of such materials. In 2000, Smith et al. experimentally demonstrated the first double-negative metamaterial, which constitutes a crucial step towards validating Pendry's predictions. ¹ These seminal contributions by Pendry and Smith established the foundation for the rapidly expanding field of metamaterials, facilitating numerous significant advancements and applications in areas such as invisibility cloaking, superlensing, and electromagnetic wave manipulation.

Figure 1.

Key historical developments of metamaterials. ¹

Following this historical development, extensive research on the design of various metamaterials has been ongoing. Consequently, metamaterials have found applications across fields, owing to their unique ability to manipulate waves and fields in unconventional ways. ⁴ In optical applications, they are utilised for superlensing to overcome the diffraction limit, beam steering, wavefront shaping, and light manipulation in optical circuits. ⁵ In the electromagnetic field, metamaterials enable the development of advanced antennas with compact designs, high gains, and multiband capabilities. ⁶ They are also pivotal in cloaking and invisibility technologies, radar systems, and stealth applications. ⁷ In the medical field, metamaterials enhance imaging techniques, such as MRI, facilitate targeted drug delivery, and improve biosensors for early disease detection due to their unique sensing properties. ⁸ Acoustic applications include noise reduction, soundproofing systems, acoustic cloaking, and enhanced ultrasound imaging. ⁹ For structural applications, metamaterials are employed in energy absorption, vibration control, impact-resistant materials, and tuneable adaptive structures.^10–12 Energy applications leverage metamaterials for efficient solar and wind energy harvesting, energy storage, wireless power transfer, and thermophotovoltaic devices.^13,14 Metamaterials are also utilised in defence and military equipment such as stealth coatings, radar-absorbing materials for aircraft and vehicles, and other military technologies. ¹⁵ Additionally, they play a significant role in quantum technologies such as quantum computing, quantum information systems, and quantum state manipulation with quantum metamaterials. ¹⁶

While there have been remarkable advancements in metamaterial research and applications in various fields, there are few comprehensive reviews synthesising the diverse range of applications and emerging trends across different domains. As such, it is necessary to undertake a comprehensive review to provide a unified understanding of the field and guide future research directions and discoveries. Within this broad spectrum, mechanical metamaterials (MMs) stand out due to their increasing significance in addressing critical challenges in areas like vibration mitigation, energy absorption, and flexible electronics and exciting properties such as high-strength-low-weight ratio, high-stiffness, and high-compression strength. However, existing reviews often focus on specific subdomains, neglecting the broader implications and interdisciplinary connections within mechanical metamaterials. Therefore, the present review examines the properties, design strategies, and applications of mechanical metamaterials, highlighting the opportunities for advancements in research and applications. The review begins by providing a general introduction to metamaterials, followed by classifications of metamaterials, focusing on mechanical metamaterials. A state-of-the-art review based on peer-reviewed data and properties of mechanical metamaterials (MMs) has been presented, followed by failure models of MMs. Finally, future perspectives and innovations in the subject are presented.

Classification of metamaterials

Based on published literature, various metamaterials have been designed and adopted for different applications. The metamaterials can be classified according to (i) properties and (ii) applications (Figure 2). Electromagnetic metamaterials exhibit unique properties such as negative refractive index, in which light bends in the opposite direction compared to conventional materials. This has potential applications in novel lenses and imaging techniques. ¹⁷ They also have invisibility cloaking properties, indicating that they can bend electromagnetic waves around an object, making it appear invisible. ¹⁸ They also exhibit high sensitivity to enhance the interaction of light with materials, leading to highly sensitive sensors for the fabrication of sensor devices. ¹⁹ Examples of such metamaterials include split-ring resonators, fishnets, and dielectric metamaterials.^20,21

Figure 2.

Classification of metamaterials according to properties and applications.

Anisotropy and dimensionality are other key properties used in classifying metamaterials. The properties in anisotropic metamaterials depend on the direction of measurement, and this is achieved by asymmetric subwavelength unit cells of the structures. It has been shown that tuneable anisotropy is desired for the adequate performance of different metamaterials, such as elliptical resonators, chiral metamaterials, and mechanical metamaterials. ²² Dimensionality is a key design parameter in metamaterials that defines how their engineered responses manifest in space. It broadly refers to the number of spatial dimensions, i.e., 1D, 2D, and 3D, in which the metamaterial's properties are engineered to operate effectively.^23,24 Dynamic and nonlinear properties describe the capability of metamaterials to react to external stimuli such as light, temperature, electric or magnetic fields, and mechanical strains. Nonlinearity describes the ability of the metamaterials to respond non-proportionally to external stimuli, and examples of such behaviour include negative stiffness, bandgap manipulation, and vibro-impact dynamics. ²⁵ These properties enhance the adaptability and reaction of the metamaterials in complex ways for advanced applications such as shock absorption, vibration isolation, acoustic cloaking, waveguides and sensors.^26,27 In electromagnetic applications, several properties of metamaterials depend on the frequency of the interacting radiation. These properties provide a powerful tool for manipulating electromagnetic radiation and have led to many exciting applications. These properties include negative refraction, electromagnetic cloaking, slow light and frequency-selective surfaces. ²⁸ Metamaterials with such properties find applications in tuneable devices, sensing, and lenses for imaging. ²⁹

In terms of applications, metamaterials are classified considering both their intended function and performance within devices under various operating conditions. As shown in Figure 2, metamaterials have electromagnetic applications in cloaking devices, super-lenses and imaging, antennas and waveguides, acoustic lenses, optical computing, energy harvesting, and meta-surfaces. ³⁰ Electromagnetic metamaterials can achieve these functionalities due to their smaller subwavelength unit cells (usually smaller than the interacting electromagnetic waves), which allow for precise control and manipulation of electrical and magnetic fields. ³¹ In acoustic and elastic wave control applications, metamaterials find applications as devices for acoustic cloaking and noise reduction, acoustic lenses, and mechanical/structural materials. ³² Other applications of metamaterials include thermal and seismic, plasmonic and quantum, as well as dynamic and adaptive systems.

The review article focuses on mechanical metamaterials, which can further be classified according to Figure 3. In this case, they can be classified according to (i) structure/geometry of the unit cell, (ii) unique mechanical properties, (iii) functionality, (iv) response to mechanical stimuli, and (v) scale level sizes of the unit cells. In terms of structure, mechanical metamaterials may contain lattice, origami/Kirigami-inspired, cellular, or hierarchical unit cells. Lattice-based metamaterials contain a repeating network of unit cells, forming a repeating arrangement of atoms analogous to a crystal (lattice) structure. The structure may be periodic or can have some degree of disorder.

Figure 3.

Classification of mechanical metamaterials.

Some of the common structures used as unit cells to design metamaterials are visually shown in Tables 1 and 2. As shown in Figure 3, unit cells are mostly geometric structures, which include hexagons, squares, rhombi, triangles, Kagome, octagons, circles, re-entrant, chiral, star-shaped, missing ribs, and others (See Table 1). ³³ The corresponding 2D lattices are represented in Table 2. Metamaterials can be tuned to achieve tailored properties through precise control of the repeating unit cell geometries, which dictate their macroscopic behaviour. The design parameters for unit cells include size, shape, and arrangement into the metamaterial structure.

Table 1.

Unit cells of the common two-dimensional (2D) lattice structures ³³ (reused with permission from Elsevier Ltd).

Table 2.

Some of the two-dimensional lattice structures ³³ (reused with permission from Elsevier Ltd).

These lattice-based metamaterials are composed of repeating unit cells arranged in a lattice pattern like a crystal structure. They are usually composed of trusses and beams and are utilised due to their design flexibility. ³⁴ The unit cells are inspired by naturally or artificially occurring geometric shapes. The specific arrangement and geometry of these unit cells give rise to extraordinary mechanical, thermal, acoustic, and electromagnetic properties that are not found in naturally occurring materials. Some of the frequently used lattice-based metamaterials and their corresponding unit cells are shown in Figure 4. These include basic cubic trusses, compound cubic trusses, 3D Kagome lattice, Bowtie honeycomb, triangular re-entrant, re-entrant star structure, gyroid, and anti-tetrarchical metamaterials. These metamaterials are recognised for their high strength-to-weight ratio, tuneable stiffness/mechanical responses, enhanced indentation resistance, and high shear modulus. ³⁵

Figure 4.

Some of the common unit cells for lattice metamaterials. Various copyrights have been obtained for the unit cells indicated in the references.^36–39

Origami/kirigami metamaterials are based on folding and cutting patterns; they are ancient paper crafts (Japanese) that have recently inspired the development of advanced materials with unique properties. Origami metamaterials are based on folding patterns, while kirigami metamaterials involve both cutting and folding. By carefully designing the folding and cutting patterns, it is possible to create materials with exciting properties, such as tuneable stiffness, negative Poisson's ratio, and configurable deformation. ⁴⁰ Typical Origami/kirigami metamaterials (and their unit cells) are shown in Figures 5 and 6. The design principles of origami and kirigami are adopted in the development of materials with excellent mechanical and electromagnetic (EM) properties, such as those for reconfigurable metamaterial absorbers. These structures possess several attractive mechanical properties, including low relative density, negative Poisson's ratio, and tuneable specific energy absorption (SEA). ⁴¹ The low relative density makes these structures lightweight, while the negative Poisson's ratio makes them expand in all directions under tension. Additionally, the tuneable specific energy absorption indicates their capacity to control energy absorption, which is favourable for various applications.

Figure 5.

The Origami and kirigami constructs for high energy absorption application (obtained via open access from MDPI ⁴² ).

Figure 6.

Geometry and development of triangular cylindrical origami (TCO shown in A and B) and multi-triangles cylindrical origami (MTCO, shown in C, D, and E) (reused under open access license from reference ⁴⁰ ).

Hierarchical metamaterials consist of multiscale structural organisation, where the geometry of the material is designed at multiple length scales (e.g., nano, micro, and macro scales). ⁴³ In these materials, the unit cells are composed of smaller unit cells that enable them to achieve multiple functionalities simultaneously or to exhibit properties that are impossible to achieve with single-level structures. An example of such a metamaterial is the stent developed from rotating grid unit cells in Figure 7. As shown, each unit cell of the stent is composed of several rotating grids assembled in a hierarchical approach. In a different work, Tian et al. ⁴⁴ studied a metamaterial fabricated from a ceramic matrix composite and exhibited a negative Poisson's ratio and high specific strength. The study involved the design of an elastic-ceramic-based composite hierarchical structure exhibiting lightweight characteristics. The authors investigated the material's mechanical properties, thermal insulation, and electromagnetic interference shielding properties. The designed composite metamaterial demonstrated a collaborative optimisation of structure and function. Hierarchical metamaterials present a range of advantages in material design, making it possible to create materials with enhanced properties and multiple functionalities. It is possible to design lightweight cellular materials with improved mechanical properties, expanding their functional applications. These materials can be engineered to exhibit a negative Poisson's ratio and high specific strength, which are desirable for various engineering applications. The hierarchical structure allows for collaborative optimisation in both structure and function, leading to materials that are not only mechanically robust but also possess tailored functional capabilities. For example, the re-entrant structure within hierarchical metamaterials can give rise to a negative Poisson's ratio response, which plays a key role in improving the material's recoverability and overall mechanical performance.

Figure 7.

Hierarchical metamaterial of the stent based on rotating grid unit cells (reused under creative commons licenses 4.0 from Nature ⁴³ ).

State-of-the-art progress in mechanical metamaterials

Negative Poisson's ratio (NPR) metamaterials

Fundamentals of NPR

Poisson's ratio (v) describes the lateral deformation of a material when subjected to axial load. It is defined as shown in equation 1:

$$\nu =-{\displaystyle \frac{{\epsilon }_{lateral}}{{\epsilon }_{axial}}}$$

(1)

Where: ε_lateral = Lateral/transverse strain and ε_axial = Axial/longitudinal strain.

Traditionally, the Poisson's ratio of materials is positive (i.e., ν > 0), meaning that they contract laterally (become narrow) when stretched. Conversely, metamaterials with a negative Poisson's ratio (NPR) expand laterally when stretched, as illustrated in Figure 8. Metamaterials with NPR are known as auxetic materials. The NPR metamaterials exhibit unique properties such as enhanced energy absorption, increased fracture resistance, and conformability to complex shapes.^45–47

Figure 8.

Demonstrating the concept of NPR in metamaterials (reused under open access license ⁴⁸ ).

Review of NPR metamaterials

The peculiar mechanical properties and diverse applications of negative Poisson's ratio (NPR) metamaterials have spurred considerable research into their design and development. Researchers have thoroughly investigated various geometries, including re-entrant honeycombs, chiral structures, missing ribs, and rotating units, to create auxetic behaviour. The rise of additive manufacturing has further propelled this field for the fabrication of intricate NPR structures with exceptional precision and customisation. The research in this field has focused on the design and performance evaluation of different unit cells for auxetic metamaterials. These structures are mostly based on geometrical shapes, and some of them have been highlighted herein.

A study by Najafi et al. ⁴⁵ has recently compared the energy absorption, compressive strength, and Young's modulus of square node anti-tetra chiral, re-entrant, and arrowhead auxetic cores against a standard honeycomb core. Quasi-static compression testing revealed a distinct ranking in compressive strength across the examined structures. The anti-tetra chiral design demonstrated the highest strength (5.947 MPa), followed in descending order by the arrowhead (5.598 MPa), re-entrant (3.140 MPa), and finally the honeycomb structure (1.391 MPa), which exhibited the lowest strength. The anti-tetra chiral structure also displayed the greatest Young's modulus (58.257 MPa), while the arrowhead structure showed the lowest value at 22.108 MPa. Throughout compression, the arrowhead structure absorbed the most energy (115.801 J), followed by the anti-tetra chiral (98.405 J) and the re-entrant (65.590 J) structures. The honeycomb structure demonstrated the least energy absorption (40.404 J) ⁴⁴ The low-velocity impact tests showed that the anti-tetra chiral structure performed best on energy absorption characteristics. The honeycomb structure had the lowest mean force during impact, while the anti-tetra chiral structure had the highest, indicating better resistance to impact. The anti-tetra chiral structure showed the lowest maximum length of stroking (MLS), meaning it absorbed energy more efficiently over a shorter deformation distance. The finite element simulations revealed good agreement with experimental results, validating the accuracy of the models. The simulations accurately predicted the structures’ deformation patterns and failure modes, particularly the layer-wise collapse observed in the honeycomb and re-entrant structures. The parametric analysis indicated that the quantity of unit cells was not a key factor influencing the anti-tetra chiral structure's performance. The slenderness ratio (height-to-width ratio) had a significant impact on performance. A slenderness ratio of 1.9 was found to be optimal for the anti-tetra chiral structure, balancing energy absorption and structural stability. The research provided valuable insights into designing and optimising auxetic sandwich structures for applications requiring lightweight, high-strength, and energy-absorbing materials.

In a similar study, Yuan et al. ⁴⁹ designed and 3D-printed auxetic metamaterials with tuneable mechanical behaviour and shape adaptability. The study focused on creating symmetric anti-chiral auxetic structures that exhibit a low Poisson's ratio of −5. As shown in Figure 9, the study evaluated the relationship between design parameters and tuneable mechanical and conformability properties of the anti-chiral metamaterial. The study revealed that Young's modulus and stress-strain response can be tuned by adjusting the central angles (θ₁ and θ₂​) of the circular arcs in the anti-chiral unit structures. As the central angle decreases, the stiffness of the metamaterial increases, leading to higher Young's modulus and greater resistance to deformation (Figure 10).

Figure 9.

Anti-chiral metamaterial developed by Yuan et al. ⁴⁹ (a) showing the parameters of the unit cells (b) metamaterials with different parameters, and (c) the 3D anti-chiral metamaterial (reused from Elsevier under creative commons CC-BY-NC-ND license).

Figure 10.

Effect of the design parameters of the anti-chiral metamaterial on the mechanical Properties ⁴⁹ (adapted from Elsevier under creative commons CC-BY-NC-ND license).

A study by Wang et al. ⁵⁰ developed a two-dimensional peanut-shaped inspired metamaterial (shown in Figure 11) and investigated the influence of design parameters on its negative Poisson's ratio. The unit cells of the lattice structure consisted of a central peanut shape connected to eight other structures (Figure 11(b)). The lattice structures were printed under different design parameters, and their Poisson's ratios were evaluated as shown in Figure 12. It was demonstrated that the structure can achieve high NPR depending on unit cell size, base material, and other cell design parameters (see Figure 11(a)). It was reported that small unit cell sizes lead to larger negative Poisson's ratios, whereas the elastic properties of the base material can be neglected. A related study on hexagonal perforated honeycomb (HPH) metamaterials was undertaken by He et al. ¹¹ and revealed that the unit cell parameters significantly affect mechanical properties and Poisson's ratio. For instance, an increase in the ratio of the elliptical perforation major axis to the minor axis of the HPH enhanced the auxetic effect of the structure (i.e., more negative Poisson's ratio).

Figure 11.

Illustrating the geometrical configuration of the peanut-inspired metamaterial developed by Wang et al. ⁵⁰ (a) The design parameters, (b) the Unit cell, and (c) the Lattice structure (reused with permission from Elsevier Ltd).

Figure 12.

Showing the printed lattice structure with PLA by Wang et al. The structure consists of 4 × 8 cells ⁵⁰ (reused with permission from Elsevier Ltd).

A huge class of metamaterials with star-shaped unit cells have been studied extensively and demonstrated to exhibit NPR properties. For instance, through finite element simulations and experiments, Mizzi et al. ⁵¹ developed a new class of auxetic metamaterials (shown in Figure 13A) with Poisson's ratios within −1 and 0. As shown in Figures 13B, C, and D, these structures exhibited a maximum of Poisson's ratio of −1 and a minimum of zero, which was confirmed through experiments and simulations. The study further revealed that the mechanical properties of star-pointed metamaterials can be tuned by altering the star perforation geometric parameters such as size, openness and number of star points. Such structures can find applications in skin grafts, stents, morphing wings, and satellite antenna components. Similar structures were designed and fabricated by Mwema et al., who demonstrated that the printing parameters influenced the mechanical performance of the structures, and negative Poisson's ratios of ∼1 were reported. ⁵² Ai and Gao ⁵³ designed four metallic metamaterials of star-shaped re-entrant planar lattice structures in another study. The study revealed that length, angle, and material combinations are suitable design parameters that influence Poisson's ratio of the structures through finite element simulation. It was further shown that metallic bi-materials fabricated via laser-based additive manufacturing can be tailored to exhibit positive, near-zero or negative Poisson's ratio without compromising the Young's modulus of such structures. These versatile properties of four-star metamaterials make them suitable for applications such as antennas and precision instruments. ⁵³ In another study, Martínez et al. ⁵⁴ introduced a novel method for designing mechanical metamaterials using Voronoi diagrams induced by star-shaped metrics. It was demonstrated that the versatility of the method to support interpolation between arbitrary metrics allows for the generation of structures with diverse mechanical properties. Readers are referred to the literature for more research on star-shaped metamaterials.^54–57

Figure 13.

Mechanical metamaterials with star-shaped pores. A. shows the 3D printed photographs of (i) 3-pointed star, (ii) 4-pointed star, and (iii) 6-pointed star metamaterials. The photographs during tensile deformation, finite element simulations and instantaneous Poisson's ratio are shown in B. for the 3-pointed star, C. for the 4-pointed star, and D. for the 6-pointed star metamaterials ⁵¹ (reused with permission from Elsevier Ltd).

Honeycomb-shaped metamaterials, which are made up of a repeating pattern of hexagonal cells, like honeycombs, are widely documented in the literature. A new class of two-dimensional (2D) second-order hierarchical honeycomb metamaterials with negative Poisson's ratio was developed by Mousanezhad et al. ¹² The structures consisted of a regular hexagonal honeycomb lattice as the base structure, the first-order hierarchy consisted of smaller hexagonal honeycombs at each vertex, and the second-order hierarchy consisted of even smaller hexagons inside the first-order hierarchical structure (Figure 14). The study concluded that hierarchical honeycomb structures exhibit auxetic behaviour due to early-stage elastic buckling, which reduces Poisson's ratio as compression increases. The research identified a critical geometric parameter (γ1) at which a switch in buckling modes occurs, leading to the lowest Poisson ratio. Introducing a higher-order hierarchy further enhanced the auxetic response. ¹² However, the main limitation was that the auxetic behaviour only emerged beyond a critical strain (∼10%), which could be reduced by optimising the geometry. In another study, Srivastava et al. ⁵⁸ investigated 2D and 3D re-entrant honeycomb metamaterials with thin strip-like beams and thin-walled tubular beams as structural elements. They explored how non-classical nonlinearities, such as Brazier's effect in thin circular tubes and geometric warping in thin strips, influence Poisson's ratio under large deformations. Their findings revealed that 2D re-entrant microstructures with thin strips exhibit a 66% improvement in auxetic behaviour under compression, while thin circular tubes enhance auxetic property under tension, with even greater improvements in 3D re-entrant geometries where the Poisson's ratio reached values as low as −4.24. The study demonstrated a clear experimental method of determining Poisson's ratio, which can be adopted in future studies. The reported method involved using a digital camera to record images of the specimen during the compression test (shown in Figure 15) at 30 frames per second. The images were analysed using MATLAB to track instantaneous axial (longitudinal) and lateral (transverse) displacements. Axial displacement was measured as the average distance change between the top and bottom plates, whereas lateral displacement was measured as the average distance change between the left and right structural columns. The measurements continued until the corners of the re-entrant unit cells came into contact, affecting further deformation.

Figure 14.

Configuration of two-dimensional honeycomb structures (regular honeycomb, first-order hierarchy, and second-order hierarchy honeycomb structures) as developed by Mousanezhad et al. ¹² (reused from Nature under open access license).

Figure 15.

Demonstrating compression test of a re-entrant honeycomb structure(thin strips) fabricated via fused filament fabrication of ABS plastic for which the elastic modulus, i.e., E = 2.2 GPa and the Poisson's ratio is ν = 0.37 as reported by Srivastava et al. ⁵⁸ (reproduced from Nature under Open Access License).

Li Yang et al. ⁵⁹ Investigated the mechanical properties of 3D re-entrant honeycomb auxetic structures (Figure 16) fabricated via additive manufacturing (Electron Beam Melting) of Ti6Al4V alloy. They developed an analytical model based on large deflection and Timoshenko beam theory to predict the structure's modulus, Poisson's ratio, and yield strength in all principal directions. Their results, validated through experiments and finite element analysis (FEA), demonstrated that geometric design parameters significantly influence auxetic behaviour, enabling tuneable mechanical properties for advanced engineering applications​. The article reported Poisson's values ranging from around −0.2 to −1.0, indicating a significant degree of auxetic behaviour. A similar study ⁶⁰ undertook large deformation analysis (both numerical and experimental) on the mechanical behaviour of re-entrant anti-trichiral honeycombs and reported Poisson's ratio values from approximately −0.3 to −1.9 in the z direction under compression, and from about −0.5 to −2.7 in the y direction. The specific value depends on the design parameters, i.e., H/L ratio and re-entrant angle θ. Compared to conventional hexagonal honeycombs, re-entrant hexagonal honeycomb structures exhibit significant anisotropy and possess greater transverse and shear stiffness (Young's moduli).^61,62

Figure 16.

Re-entrant honeycomb structure showing (a) 2D configuration, (b) 3D lattice, and (c) unit cell as reported by Li Yang et al. ⁵⁹

Wanniarachchi and colleagues have published two notable articles on the 3D printing of cobalt-chromium-molybdenum (CoCrMo) metamaterials for biomedical applications based on structures related to re-entrant/honeycomb morphologies.^63,64 They proposed a framework for developing CoCrMo auxetic meta-biomaterials suitable for load-bearing bone scaffolds, investigating their potential for near-zero and highly negative Poisson's ratios. ⁶³ In this case, five different CoCrMo auxetic meta-biomaterial scaffold designs (AX1-AX5) were manufactured via laser-powder bed fusion (L-PBF). Characterisations included uniaxial compression testing to evaluate mechanical performance and failure modes, scanning electron microscopy (SEM) to analyse print quality and dimensional accuracy, and a multi-criteria decision-making procedure combining the analytic hierarchy process (AHP) and the technique for order of preference by similarity to ideal solution (TOPSIS) to identify optimal architectures. The research reported Poisson's ratios ranging from −0.1 to −0.24 and porosities between 73% and 82%. Building on this foundation, the subsequent work further details the development of a CoCrMo meta-biomaterial structure designed to facilitate personalised stiffness-matching while exhibiting near-zero auxeticity. ⁶⁴ This later work introduced a novel surrogate model, developed through prototype testing and numerical modelling, capable of predicting key mechanical properties such as porosity, yield strength, elastic modulus, and negative Poisson's ratio of the meta-biomaterial. The findings corroborated the potential for CoCrMo meta-biomaterials, demonstrating successful achievement of near-zero auxetic performance (≤0.037) and targeted stiffness-matching, with parametric analysis showing Poisson's ratios between −0.02 and −0.08 and porosities between 73.63% and 81.38%. These references collectively underscore the advancement in designing and characterising additively manufactured auxetic CoCrMo meta-biomaterials for critical biomedical applications.

Another class of NPR metamaterials is that developed from rotating (semi-) rigid structures using squares, rectangles, triangles, rhombi, and parallelograms. ⁶⁵ Sorrentino et al. reported that the geometric shapes are connected via hinges, so they can rotate on loading and expand or contract as desired. Study ⁶⁶ explored the mechanical properties in the axial and non-axial directions of auxetic metamaterials, focusing on a novel design inspired by rotating squares. The researchers designed and optimised a combined rotating/chiral architecture with hollow rotating units using finite element analysis to reduce stress concentrations and enhance strain tolerance. They investigated the impact of incorporating ligaments with non-uniform thickness and spline fillet profiles on mechanical properties such as Poisson's ratio. The effective Poisson's ratio of the simulated structures was observed to be consistent during on-axis stretching, reaching a minimum of around −0.96 in the case of length of the ligament (l_a) = 1.0 mm and location (P¹_y) = 1.0 mm while slightly increasing until −0.94 when both l_a and P¹_y rise to 2.6 mm and 1.8 mm respectively. The study fabricated a prototype using 3D printing, demonstrating a strong agreement between experimental results and numerical predictions, thereby validating the design approach. Another study reported the in-plane Poisson's ratio for three types of material models: a purely rotating squares model, a model with sub-units exhibiting relative motion, and a model with sub-units originating from a single square undergoing relative motion. ⁶⁷ The study found that the proposed metamaterials demonstrate a transition in Poisson's ratio behaviour, characterised either by discontinuity or continuous but non-differentiable Poisson's ratio at the transitory state between deformation mechanisms. The structures exhibited a combined rotating/chiral architecture with enhanced mechanical properties, achieving a Poisson's ratio of about 0.94. The study explored modified auxetic structures derived from rigid rotating squares, focusing on a novel connection method using axes of rotation on the square surfaces. This design aimed to create a reliable system capable of complete surface coverage and pore formation. The research successfully produced a new 2D auxetic metamaterial, demonstrating a Poisson's ratio of −1, independent of the mechanical properties of the rigid squares. The theoretical relationships were validated via physical models, showcasing the structure's potential for applications in adjustable filtration systems. Geometric parameters (such as length and angular dimensions) of the rotating square structures significantly influence the Poisson's ratio and mechanical behaviour of such metamaterials; an example is shown in Figure 17. ⁶⁸

Figure 17.

Demonstrating the relationship between the geometrical parameters and Poisson's ratio of rotating square structures with holes ⁶⁸ (adapted from MDPI as open access under creative commons attribution (CC BY) license).

Another set of auxetic metamaterials includes chiral structures, which are characterised by a specific arrangement of objects leading to asymmetry, and they cannot be superimposed into their mirror image. An example of a chiral structure consists of a central cylinder encapsulated in tangentially attached ligaments, as shown in Figure 18. The focus of research in these structures is on optimising the auxetic properties, and researchers are developing new structures for better performance. For example, a recent paper introduced an optimised lozenge-chiral auxetic metamaterial (oLCAM) designed to balance auxetic behaviour and stiffness while achieving tuneable properties. ⁶⁹ The study compared the Poisson's ratio and stiffness of the oLCAM with a conventional lozenge auxetic metamaterial (cLAM) through experiments and numerical simulations. The study demonstrated the ability to optimise the Poisson's ratio and stiffness of the oLCAM, showcasing its potential for applications where both auxetic behaviour and mechanical strength are required (Figure 19).

Figure 18.

An example of a hexachiral structure consisting of a central cylinder and tangentially embedded ligaments, which cannot be superimposed on its mirror image. The image shows the deformation of the structure in the y-direction. ⁶⁵ (reused from RSC publishing under Open Access BY-NC license).

Figure 19.

The geometry of chiral structures developed by Hou et al. ⁶⁹ (a) cLAM, (b) LCAM-2R, ³⁸ (c) oLCAM, (d) unit cell of each structure, and (e) detail configuration of the oLCAM's unit cell. The Poisson's ratios of the different structures are also shown (reused from Elsevier under the Creative Commons CC-BY-NC-ND license).

Another chiral structure exhibiting auxetic properties is shown in Figure 20, ⁷⁰ which was studied for mechanical properties using numerical methods under different loading conditions, including uniaxial loading and torsion with superimposed uniaxial loading, biaxial loading, triaxial loading, and hydrostatic loading. The auxetic behaviour of the structure is gradually lost as torsion is applied (Figure 21). At higher compressive loads, the Poisson's ratio is higher, whereas at higher tensile loads, the Poisson's ratio is lower (Figure 21).

Figure 20.

The geometry of the chiral auxetic structure was investigated numerically by Kose et al. ⁷⁰ (reused from John Wiley and Sons through open access under the creative commons CC-BY-NC-ND license).

Figure 21.

Variation of Poisson's ratio with axial strain during (a) compression and (b) tensile uniaxial loading as reported by Kose et al. ⁷⁰ (reused with permission from John Wiley and Sons through open access under the creative commons CC-BY-NC-ND license).

Li et al. ⁷¹ demonstrated a method for designing 3D chiral and achiral mechanical metamaterials using spherical tiling. These metamaterials were fabricated through multi-material 3D printing, resulting in cellular and composite designs incorporating a soft matrix. To evaluate their mechanical behaviour, uniaxial compression experiments were conducted. The primary results indicated that the 3D chiral designs exhibited a coupled compression-twisting behaviour, whereas the 3D achiral designs displayed auxetic behaviour. Furthermore, the metamaterials’ effective mechanical properties and deformation mechanisms could be adjusted by modifying the chiral geometry and material combinations. Chen et al. ⁷² demonstrated a topology optimisation approach to design materials for tubes and beams that exhibit twist deformation under axial strain. The metamaterials used were cellular and composite materials with microstructures designed to maximise the twist angle of the structure. The tests involved numerical simulations using the bi-directional evolutionary structural optimisation (BESO) method. The main results showed that the optimised microstructures achieved desirable twist chirality in tubes and beams and that the twist chirality depends on the size of the material unit cell. In a similar work, Fu et al. ⁷³ demonstrated the design of a novel three-dimensional chiral honeycomb material, assembled orthogonally from a two-dimensional chiral honeycomb with four ligaments. Beam theory was employed to derive analytical expressions for Young's modulus and Poisson's ratio. The calculated results obtained using these formulae showed a good correlation with those generated by the finite element method. The study indicated that, at a macroscopic scale, the honeycomb material displays isotropic behaviour, exhibiting a Poisson's ratio nearing −1. Furthermore, the impact of geometric parameters on the overall elastic properties was investigated. It was shown that the equivalent Young's modulus is proportional to the biquadrate of the ratio of thickness to length (t/l), indicating that increasing the ligament's thickness relative to its length significantly increases the material's stiffness. It was also shown that the angle between the ligament and the imaginary line connecting node centres (θ) influences both Young's modulus and Poisson's ratio. Furthermore, the aspect ratio of the ligament (β, or h/t), which is the ratio of the ligament's depth to its thickness, also plays a role in determining the material's elastic properties. Drawing inspiration from the geometrical arrangement of left (LH) and right-handed (RH) climbing-towel gourd tendrils, Wu et al. ⁷⁴ proposed an innovative chiral architected cylindrical tube that converts axial compression into angular rotation.

Based on the foregoing literature, chiral metamaterials exhibit several interesting properties. Firstly, they lack mirror symmetry. This chirality gives rise to unique behaviours, such as a distinctive interaction with circularly polarised light and the ability to enhance optical chirality. Furthermore, chiral metamaterials can display unusual mechanical responses. For instance, some 3D chiral designs demonstrate a coupled compression-twisting behaviour, and their effective mechanical properties can be adjusted by altering their geometry and material composition. Additionally, these metamaterials can be designed to possess a negative Poisson's ratio (auxetic behaviour). Chiral metamaterials also have potential in sensing applications, particularly for detecting chiral molecules.

Energy absorption metamaterials

Fundamentals of energy absorption in metamaterials

In applications demanding impact mitigation, crashworthiness, vibration damping, and acoustic shielding, structures with high energy absorption properties are desired. This demand, coupled with lightweight requirements in aerospace, automotive, and energy sectors, places metamaterials at the centre of research. As such, extensive efforts are being made to develop metamaterials for high-energy absorption.^75,76 The energy absorption capacity (EA) can be obtained by integrating the load-displacement curve (Figure 22) up to the start of the densification region (equation 2):

$$EA=\underset{0}{\overset{\delta }{\int }}Fd\delta$$

(2)

Figure 22.

Force vs displacement curve of a typical metamaterial during a compression test. As shown, there are three regions of the graph, i.e., elastic region, plateau stage, and densification region. (Adapted with permission from Elsevier Ltd under open access Creative Commons CC-BY-NC-ND license ⁷⁹ ).

Where $\delta$ is the deformation at the beginning of the densification of the metamaterial structure, and F is the compression force as a function of the compression distance (Figure 22).

Specific energy absorption (SEA) is used to quantify the energy absorption properties of metamaterials as in equation 3.

$$SEA={\displaystyle \frac{EA}{m}={\displaystyle \frac{\int Fd\delta }{m}}}$$

(3)

Where m is the mass of the structure, and F is the applied force.

The rigid-plastic collapse model for honeycomb and lattices, defined as the mean crushing stress ${\sigma }_{mean}$ is estimated as shown in equation 4.

$${\sigma }_{mean}=C.{\sigma }_y{\left({\displaystyle \frac{\rho }{{\rho }_s}}\right)}^n$$

(4)

Where C and n are empirical constants, ${\sigma }_y$ is the yield strength of the base material, $\rho$ is the relative density of the structure, and ${\rho }_s$ is the density of the solid material. For honeycomb structures under axial compression, the stress is given by ${\sigma }_{mean}=\frac{4}{9}{\sigma }_y\left(\frac{t}{l}\right)$ , where t is the cell wall thickness and l is the cell size.

For high-strain rate impacts, metamaterials behave like shock absorbers. The Hugoniot equation describes the pressure-density relation: $P={\rho }_o{c}_o^2\left(1-\frac{{\rho }_o}{\rho }\right)$ , where $P_o$ is the initial pressure, ${\rho }_o$ and $\rho$ are the initial and final densities, and $c_o$ is the bulk speed of sound in the material.

The above theories have been applied to different metamaterials to evaluate and optimise them for energy absorption applications. The parameters can be obtained from load-displacement curves obtained from a uniaxial compression test. Generally, the load-displacement curve of a metamaterial consists of three regions: the elastic, plateau, and densification regions (Figure 22). ⁵² The energy absorbing efficiency ( $\eta$ ) is the quotient of EA and the maximum compression force ( $F_{max}$ ). It is calculated using the relationship in equation 5.

$$\eta ={\displaystyle \frac{EA}{F_{max}}={\displaystyle \frac{{\int }_0^{\delta }Fd\delta }{F_{max}}}}$$

(5)

SEA is a crucial parameter for measuring the energy absorption efficiency of materials under impact loading. SEA is widely used in designing lightweight and high-performance structures, particularly in aerospace and automotive industries, where energy dissipation must be maximised while keeping weight to a minimum. ⁷⁷ In crashworthiness applications, materials with high SEA improve passenger safety by effectively absorbing kinetic energy during collisions, making them essential for vehicle crumple zones and impact-resistant aerospace components. Similarly, in defence and military applications, SEA is used to assess the effectiveness of blast-resistant and ballistic protection materials, ensuring that shock waves and impact forces are dissipated efficiently. ⁷⁸ In biomechanics, SEA helps in designing protective gear, including helmets and body armour, ensuring maximum impact mitigation with minimal weight. Structural and seismic engineering also benefit from high SEA materials, as they contribute to earthquake-resistant designs by efficiently absorbing seismic energy and reducing structural damage.

Other parameters used in the measurement of energy include mean crushing force (MCF) and crushing force efficiency, given according to equation 6. ⁷⁸

$$MCF={\displaystyle \frac{EA}{{\delta }_{maz}}\mathrm{and}CFE={\displaystyle \frac{MCF}{IPF}}}$$

(6)

Where IPF is the initial peak force.

Progress in research on energy absorption metamaterials

Extensive literature is available on the various metamaterials for energy absorption applications. As summarised in Table 3, EA and SEA are the most used parameters in energy absorption measurement for metamaterials and the raw data is mostly obtained from a compression experiment and backed by numerical simulations. Habib et al. ⁸⁰ studied the energy absorption of metamaterials consisting of six different polymeric lattice unit cells, namely circular, octagonal, strengthened octagonal, Kelvin, Rhombicuboctahedron (RO), and cubic structures. Each metamaterial had a relative density of 0.15 and was fabricated via Multi Jet Fusion technology. The structures were characterised via quasi-static energy absorption property using finite element analysis (FEA) and experiments. It was reported that metamaterials with octagonal unit cells exhibited the optimum energy absorption properties. In a related study, Zhao et al. ⁸¹ investigated energy absorption characteristics of body-centred cubic (BCC) lattice structures and triply periodic minimal surface (TPMS) based BCC lattice structures (see Figure 23). TPMS-based BCC lattice structures exhibited the highest energy absorption. Similarly, Zhang et al. ³⁹ explored the mechanical behaviour and energy absorption capabilities of three types of TPMS sheet lattices, Primitive, Diamond, and Gyroid, produced from 316L stainless steel using selective laser melting (SLM) under compressive forces. It was shown that diamond-type TPMS sheet structures exhibited the highest energy absorption, whereas the lowest energy absorption was reported in BCC lattices.

Figure 23.

Showing BCC unit cell and corresponding CAD model and fabricated metamaterial, and BCC-TPMS unit cell and corresponding CAD model and AM metamaterial. A bar graph of energy absorption vs volume fraction of the BCC and BCC-TPMS ⁸¹ (Adapted from MDPI under Open Access Creative Commons Attribution (CC BY) license).

Figure 24.

Energy absorption characteristics obtained during quasi-static compression testing of novel meta-structures developed by Almesmari et al. ⁸⁷ (Adapted from Taylor & Francis as open access under the Creative Commons CC BY license).

Figure 25.

(a) Bend-dominated honeycomb (BHC) (b) stretch-dominated honeycomb unit cells for the hierarchical spherical hollow structures described in reference ⁷⁵ (c) specific strength and SEA and (d) average plateau stress and densification strain for various configurations of the metamaterials (adapted with permission from Elsevier Ltd).

Figure 26.

(a) Euplectella aspergillum-inspired structures with four multi-cell tube configurations (two perspectives)-simple square (SS), face-centred square (FCS), and modified face-centred cubic (MFCS) (b) initial peak crushing force, (c) mean crushing force, (d) SEA, and crushing force efficiency for the four structures (adapted from ⁷⁹ under Open Access License).

Figure 27.

(a) Gyroid unit cell at different views, (b) helical support structure parameters, and (c) reinforced-gyroid unit cells at different views (obtained from ⁹¹ under an open access license).

Figure 28.

Descriptions of the four re-entrant structures benchmark re-entrant (BR), single symmetry-broken re-entrant (SBR), double symmetry-broken re-entrant (DBR) and Hybrid symmetry-broken re-entrant (HBR) as reported by Montazeri et al. ⁹³ (reproduced with permission from Elsevier Ltd).

Figure 29.

Showing (A) conventional BCC and FCC and corresponding ribbed unit cells (B) relative densities for (i) conventional and (ii) ribbed BCC and FCC, (C) 316L fabricated metamaterials and (D) front view of the respective unit cells ⁹⁵ (reproduced with permission from Taylor & Francis under Open Access).

Table 3.

Some of the structures and their energy absorption characteristics as reported in the published literature.

Reference/ Aim of the Work	Structure description	Energy absorption behaviour
Almesmari et al. (2024). 87 The article developed and modelled two new topologies, for Compression Internal Energy Dissipation (CompIED) and Shear Compression (ShRComp), respectively. The energy absorption and impact resistance of the two topologies were also explored.	The unit cells can be described as (1) a simplified human-like torso with four protruding limbs (CompIED) and (2) two interconnected human-like torso (ShRComp). See Figure 24.	The EA was shown to increase with the relative densities of the novel structures. At a relative density of 0.5, EA of about 1.4 MJ/m3 (Figure 23). The maximum EA was obtained as 96.1 MJ/m3).
Chikkanna et al. (2025). 78 This research investigated how a newly designed re-entrant diamond auxetic (RDA) metamaterial behaves under dynamic crushing and its capacity to absorb energy. The study also involved optimising the structural designs to enhance these dynamic crushing and energy absorption properties.	The unit cell is a re-entrant diamond auxetic (RDA) metamaterial. The unit cell design is meant to mitigate bending mode deformation.	The optimised RDA structure absorbed 2669 J of energy with a specific energy absorption (SEA) of 10.5 J/g. These values represent improvements of 71% in energy absorption and 22% in SEA when compared to traditional re-entrant auxetic metamaterials. Furthermore, the study showed enhancements of 230% in energy absorption and 42% in SEA over hexagonal honeycomb structures of equivalent unit cell size.
Gao et al. (2023) 88 The study sought to create a group of rubbery and recoverable composite structures inspired by twisting biological fibres, designed to maintain their mechanical properties without significant degradation. The specific energy absorption of these structures was also examined.	The structure is a twisting fibre composite structure with different fibrous morphologies.	A machine learning model was successfully created to forecast the SEA of the structures. The maximum value of SEA obtained was about 3000 MJ/g.
Gao et al. (2022) 75 The study designed hierarchical spherical hollow lattice structures with honeycomb morphologies for bending and stretch-dominated deformation. Their energy absorption properties were reported with other mechanical properties.	Two hierarchical spherical hollow structures (SHS) (1) containing bend-dominated honeycomb (BHC) and (2) stretch-dominated honeycomb (SHC) unit cells. See Figure 25.	Based on the various configurations presented, the hybrid metamaterial of SHC and BHC exhibited the highest SEA at 15.25 J/g. See Figure 25.
Ha et al. (2022) 89 This research explored how novel bio-inspired hierarchical circular honeycomb (BHCH) structures, designed to resemble the hierarchical organisation of wood, absorb energy. The study examined these energy absorption characteristics to the structures’ geometric parameters and the materials used in their construction.	The study proposed BHCH structures, which consisted of a periodic arrangement of cylindrical units arranged in a square-packed configuration and with additional internal small cylinders to form a honeycomb arrangement as inspired by wood structure. As a control sample, circular honeycomb (CH) structures were also investigated.	The study reported SEA values of 6.07 kJ/kg and 8.82 kJ/kg for CH and BHCH respectively.
Wang et al. (2023) 79 The study explored the energy absorption and crashworthiness of a novel multi-cell bio (Euplectella aspergillum)- inspired design for thin-walled tubular structures with modified face-centred cubic (MFCS) cross-sections (Figure 26).	Euplectella aspergillum-inspired structures with four multi-cell tube configurations-simple square (SS), face-centred square (FCS), simple square-face-centred square (SS-FCS), and modified face-centred cubic (MFCS) (Figure 26).	The modified face-centred cubic (MFCS) structure revealed a maximum SEA of 72 J/g.
Yang et al. (2018) 90 The study optimised a composite energy-absorbing structure consisting of thin-walled steel tubes and a honeycomb support (internal) metamaterial structure.	The structure consists of a truss-like framework with a honeycomb internal support metamaterial.	The optimised structure obtained an energy absorption of 2440 kJ.
Tilley et al. (2024) 91 The study employed compression tests to assess the compression modulus and energy absorption of the gyroid structures.	The research developed two structures: (1) Gyroid unit cell and (2) helical-reinforced gyroid unit cell metamaterials. See Figure 27.	The SEA increased with the relative density of the structures. The maximum SEA for the gyroid was 2600 kJ/m3, whereas for the reinforced gyroid was reported to be 2800 kJ/m3.
Li et al. (2024) 92 The study aimed to develop multifunctional bioinspired architected metamaterials (MBAMs) for sound-absorbing and mechanical applications.	The MBAM structure mimicked the cuttlebone of a cuttlefish. It consisted of multilayered vertical and cambered cell walls. The structure was fabricated from high-strength Ti6Al4V through the selective laser melting (SLM) process.	A maximum SEA of 50.7 J/g was reported for samples with a bottom (curved shape) profile length of 1.5. This value indicated a 558.4% increase over the straight wall design.
Montazeri et al.(2024). 93 The study compared the energy absorption properties of conventional re-entrant structures, single symmetry-broken and double symmetry-broken re-entrant structures.	The structures investigated were conventional re-entrant structures, single symmetry-broken and double symmetry-broken re-entrant structures (Figure 28)	The SEA values obtained were: 77.2 J/Kg, 157.3 J/Kg, 120.8 J/Kg, and 133.1 J/Kg for BR, SBR, DBR, and HBR, respectively.
Yuan, Chua, & Zhou (2018). 94 The article reported on the energy absorption of 3D metamaterials of auxetic composites manufactured via laser sintering of carbon nanotubes reinforced nanocomposites.	The 3D metamaterial consisted of body-centred cubic (BCC) with 6 and 12holes as the unit cells. The unit cells were then arranged in a cube to form a 3D mechanical metamaterial.	The optimised structures exhibited a SEA of 20.42 J/g and an energy absorption capacity of 6.29 MJ/m3.
Wang et al. (2024). 95 The authors reported on the mechanical properties of 316L fabricated metamaterials consisting of ribbed FCC and BCC unit cells. The results were compared with those of the conventional BCC and FCC unit cells.	The metamaterials were composed of unit cells of BCC and FCC with ribbed truss-lattice structures denoted as BCCR and FCCR (Figure 29)	There was an increase in SEA from 5.5 J/g to 9 J/g (an increase of 109%) for BCC to BCCR. The SEA for FCC was 8.9 J/g, whereas for FCCR was reported to be 17 J/g, which indicates a 94% increment in SEA.

DBR: Double symmetry-broken re-entrant.

An article by Sharma and Hiremath in 2022 titled, “Additively manufactured mechanical metamaterials based on triply periodic minimal surfaces: Performance, challenges, and application”, undertook an extensive review of the triply periodic minimal surface (TPMS)-based mechanical metamaterials, including a discussion of energy absorption. ⁸² It was demonstrated that for a cellular structure, the energy absorption capability depends on the relative density, geometry of the unit cell, base material property, and loading rate. Similar results were demonstrated by Sadeghi et al. ⁸³ for TPMS for shape memory applications. Of the TPMS structures discussed in the review article, the bending-dominated ones (e.g., sheet-based TPMS lattices) have a larger stress plateau under compression, which means they can be used for energy absorption applications. The energy absorption capacity can be affected by the design parameters of the TPMS, such as the type of unit cell (primitive (P), gyroid (G), or gyroid (G)), the sheet thickness, and the number of layers. The maximum specific energy absorption was achieved by sandwich panels incorporating a P-TPMS core; it was demonstrated that the energy absorption efficiency decreased as the layer thickness and the quantity of layers increased. ⁸²

Hierarchical designs offer a key advantage by enabling the manipulation of a material's collapse mechanism, which leads to improvements in both energy absorption and compressive strength. ⁸⁴ For instance, the introduction of hierarchical configurations, such as hexachiral auxetic subordinate cells, can extend plastic deformation and maintain auxetic behaviour at larger strains. This results in a more stable and prolonged post-yield plateau in the compressive stress-strain curve, thereby significantly enhancing energy absorption. Furthermore, hierarchical metamaterials can be designed to achieve a higher specific energy absorption (SEA) compared to conventional metamaterials. In the context of sandwich structures, hierarchical designs have demonstrated an improved capacity for energy absorption compared to regular honeycomb sandwich panels. ⁸⁵ Moreover, hierarchical metamaterials can be tailored for multi-functional applications, such as combining energy absorption with vibration insulation. ⁸⁶ This is accomplished by incorporating additional structural elements, like square unit cells or mass inclusions, into the hierarchical design, which enhances both energy absorption and vibration insulation capabilities. Table 4 provides a comparison of different hierarchical metamaterials with the respective traditional metamaterials in terms of energy absorption from the literature. From the table, it is evident that hierarchical metamaterials exhibit better energy absorption characteristics.

Table 4.

A summary of literature comparing the crashworthiness performance of conventional and hierarchical honeycombs. (The table was reused under the Creative Commons CC-BY-NC-ND license). ⁹⁶

Material	Type of loading	Structure	Control	Major finding	Ref.
Verowhite	Quasi-static	Square hierarchical honeycomb (SHH)	Regular square honeycomb (RSH)	54.4%, 117.3%, and 126.7% increase in SEA	97
AA6061-O	Quasi-static	Second-order hierarchical honeycomb	Regular honeycomb and aluminium foam	2.63 and 4.16 greater plateau stress	98
AA3003H18	Dynamic	Horseshoe honeycomb	Triangular, square, hexagonal, and Kagome honeycomb	37%, 38.4%, 6.5%, and 12.4% increase in SEA	99
Al6061-T5	Quasi-static	Bionic honeycomb tubular nested structure (BHTMS)	–	29.3 J/g	100
AA6060-T4	Quasi-static and dynamic	Self-similar hierarchical honeycomb	Regular honeycomb	Higher plateau stress	101
Al6063-T5	Quasi-static	1st, 2nd, 3rd and 4th order hierarchical honeycomb	Regular honeycomb (zero order)	71%, 114%, 201%, and 309% in MCF	102
AA3030-H19	Dynamic	1st, 2nd, 2nd, 3rd hexagonal, Kagome and triangular honeycomb	Regular honeycomb	Twice greater EA	103
AA6060-T4	Dynamic	1st order and 2nd order hierarchical spider-web	Regular honeycomb (0 order)	62.1% and 82.4% higher SEA	104
Solid steel Q235	Quasi-static	Hierarchical lattice tube (HLT) and super hierarchical lattice tube (SHLT)	Single-cell tube (ST)	108% and 176% increase in MCF	105
AA3003	Quasi-static	Ripplecomb (corrugated triangular honeycomb)	Regular triangular honeycomb	73.8% increase in SEA	106

Ebrahimi et al. ¹⁰⁷ reported the development and fabrication of four innovative hybrid metamaterial structures which incorporate honeycomb, re-entrant, and star-shaped unit cells, re-entrant-star type 1 (RS1), honeycomb-star type 1 (HS1), re-entrant-star type 2 (RS2), honeycomb-star type 2 (HS2) (Figure 30). Subsequently, the in-plane energy absorption capacity and uniaxial compressive behaviour of the newly designed structures were compared through finite element simulations and experimental methods. The key results reported by Ebrahimi et al. ¹⁰⁷ were: (i) Regarding mechanical performance, the RS1 structure outperformed the others. Its compressive strength was about 1.68 times that of HS1 and three times that of RS2. Moreover, the plateau stress for RS1 was 1.86 times higher than for HS1. (ii) The RS1 structure also showed superior energy absorption, absorbing over twice the energy of RS2 and roughly 1.64 times the energy of HS2. Its specific energy absorption was also approximately 2.47 times that of RS2. (iii) As illustrated in the stress-strain curve (Figure 30), cellular materials typically display an extended plateau region. The RS1 structure exhibited this characteristic, showing a stable curve without significant oscillations upon material failure, unlike the RS2 structure. (iv) The failure sequence in the HS1 structure began with the star-shaped cells collapsing, and then the honeycomb cells. This controlled failure mode could be advantageous for directing where damage occurs in different applications.

Figure 30.

3D printed structures of (a) re-entrant-star type 1 (RS1), (b) honeycomb-star type 1(HS1), (c) re-entrant-star type 2 (RS2), (d) honeycomb-star type 2 (HS2). The unit cells are shown as insets on the 3D-printed pictures. The stress-strain curves during in-plane compression tests are also shown on the right-hand side of the figure 107 (Adapted from Elsevier under Open Access Creative Commons CC-BY license).

It has been reported that materials consisting of re-entrant honeycomb structures (hybrid) have attractive auxetic properties compared to traditional honeycomb. ¹⁰⁸ Such structures are lighter, exhibit higher SEA, and have higher tunability. Dong et al. ¹⁰⁸ presented a metallic auxetic re-entrant honeycomb through selective laser melting (SLM). Investigating the effect of cell-wall thickness on energy absorption efficiency, the study determined that elastic strain energy plays a crucial role in the total energy absorption of thin-walled honeycombs and should not be overlooked. In a related study, Xu et al. ¹⁰⁹ investigated the in-plane mechanical properties and energy absorption capability of the AuxHex structures by using theoretical, finite element simulation and experimental methods. The structure (AuxHex) combines hexagonal honeycomb and auxetic (re-entrant hexagonal honeycomb) cells. In this study, Xu et al. ¹⁰⁹ successfully fabricated a novel AuxHex structure using the selective laser sintering (SLS) technique. The AuxHex structure demonstrated superior mechanical properties compared to traditional honeycomb structures. Specifically, the modulus and collapse stress of the AuxHex structure were higher than 16% of the conventional honeycomb structure. The SEA of the AuxHex structure was also significantly higher, with an increase of over 38% observed in the x-direction. The theoretical predictions made by the authors were consistent with the results obtained from finite element analysis (FEA) and experiments. A similar structure known as a novel square re-entrant honeycomb (SRH) by incorporating square unit cells into a traditional re-entrant honeycomb (RH) was developed by Ma et al. in 2023. ⁸⁶ This study theoretically and numerically examined the dynamic crushing behaviour of SRH structures, encompassing their deformation patterns, crushing stress, energy absorption capacity, and vibration isolation capability. The findings revealed that SRHs exhibited greater plateau stress and specific energy absorption (SEA) compared to RH structures across various crushing speeds. Tao et al. ¹¹⁰ proposed a comparable structure in 2022, which involved adding small re-entrant structural units to the joints of conventional re-entrant designs. However, Tao et al. ¹¹⁰ explored the vibrational properties and bandgap tunability of the re-entrant honeycomb, and therefore the design by Ma et al. ⁸⁶ advanced the application of such structures.

Dong et al. ¹⁰⁸ studied the compression behaviour of graded metallic auxetic re-entrant honeycomb. Two types of graded auxetic re-entrant hexagonal honeycombs were prepared by selective laser melting (SLM) and their mechanical properties were investigated through quasi-static compression experiments and finite element (FE) simulations. The effects of gradient distribution on the deformation mode, crushing stress, Poisson's ratio and energy dissipation characteristics were analysed. The authors reported that the graded distribution had a great effect on the energy dissipation characteristics of the graded auxetic honeycomb. Up to the densification of the graded layer with the thickest cell walls, the unidirectionally-graded auxetic honeycomb (UGAH) exhibited less energy dissipation compared to the bidirectionally-graded auxetic honeycomb. Distefano and Epasto ¹¹¹ investigated the effect of density grading on the energy absorption characteristics of Triply Arranged Octagonal Rings (TAOR) lattices (Figure 31) manufactured from Ti6Al4V through the Electron Beam Melting (EBM) process. The results of the study indicated that the density gradient had an impact on the mechanical behaviour of the TAOR lattice structures, with the compressive strength decreasing as the density gradient increased (Figure 31). The elastic modulus of the TAOR lattices was found to be in the range of human bone, indicating that they have potential for use in biomedical applications.

Figure 31.

Showing the unit cells, 3D-printed metamaterials of TAOR, and variation of SEA with their relative densities ¹¹¹ (reproduced with permission from Elsevier under the Open Access Creative Commons CC-BY license).

Failure of mechanical metamaterials

Fracture toughness

Fundamentals

Optimising performance and practical application of metamaterials requires a deeper understanding of their failure modes. Unlike traditional materials, the failure of metamaterials is primarily dictated by their architecture, and as such, their failure is interesting and complex. The complexity of the metamaterial failure is due to the complex interplay among the structural design, material properties, and applied loading. In studying the failure of metamaterials, fracture toughness is an important parameter as detailed in fracture mechanics, but modified to account for the geometrical influence of the metamaterials. ¹¹²

For Mode I fracture toughness (crack opening mode), the fracture toughness, K_IC for a metamaterial subjected to tensile loading is given by $K_{IC}=Y{\sigma }_c\sqrt{\pi a}$ , where Y is the geometry factor, which depends on the crack shape and boundary conditions, ${\sigma }_c$ is the critical stress at failure, and a is the crack length. For Mode II fracture toughness (shear mode), the fracture toughness, K_IIC is given by $K_{IIC}=Y^{\mathrm{\prime }}{\tau }_c\sqrt{\pi a}$ where $Y^{\mathrm{\prime }}$ and ${\tau }_c$ represent the geometry correction factor for shear loading and the critical shear stress, respectively. The energy-based fracture toughness is studied through strain energy release rate (G_C) and is adopted in fracture analysis of soft metamaterials and computed as follows; $G_C=\frac{K_C^2}{E*}$ , where K_C is the fracture toughness and E* is the effective modulus of the metamaterial.

In elastic-plastic fracture mechanics, the J-integral characterises the energy release rate, defined as the strain energy dissipated per unit area of the fracture surface, in materials exhibiting nonlinear elastic-plastic deformation. It has been used in fracture toughness measurement for metamaterials with various modifications. The general J-integral is expressed as a contour integral around the crack tip according to equation 7:

$$J=\int \mathrm{\Gamma }(W\delta 1j-\sigma ij\mathrm{\partial }ui\mathrm{\partial }x1)njd\mathrm{\Gamma }J=\underset{\mathrm{\Gamma }}{\int }\left(W{\delta }_{1j}-{\sigma }_{ij}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_1}}\right)n_j$$

(7)

where:

• $W=\int 0\epsilon ij\sigma ijd\epsilon ijW=\underset{0}{\overset{{\epsilon }_{ij}}{\int }}{\sigma }_{ij}d{\epsilon }_{ij}$ is the strain energy densit y

• $\sigma ij{\sigma }_{ij}and\epsilon ij{\epsilon }_{ij}$ are the stress and strain tensors

• $ui{u}_i$ are the displacement component s

• $x1x_1$ is the coordinate in the crack propagation direction

• $nj{n}_j$ is the outward normal to the contour $\mathrm{\Gamma }\mathrm{\Gamma }$

• $d\mathrm{\Gamma }d\mathrm{\Gamma }$ is the infinitesimal contour length

Case studies

The fracture toughness in metamaterials depends on the connectivity of the lattice (which determines whether it is stretch or bending-dominated), relative density, size of the unit cell, and configuration of the ligaments. ¹¹³ In 2023, Choukir and Singh used finite element simulations, employing both J-integral and stress intensity factors, to analyse the relationship between the structure (topology) and resistance to fracture (fracture toughness) in various types of periodic cellular materials (illustrated in Figure 32). ¹¹⁴ The authors reported that sheet lattices are the toughest at higher densities, and at lower relative densities, sheet lattices exhibit a faster decline in performance because of plate buckling, a failure mode that affects them more significantly than strut lattices in mode 1. The authors also found that in low-density scenarios, TPMS sheet lattices offer a more advantageous option than plate lattices regarding fracture toughness, as their scaling with density is nearly linear. Furthermore, it was found that the paths of crack growth in 3D cellular materials are determined by the fracture geometry. However, for centre cracks under mode I load, 3D lattices exhibit a straight crack path, regardless of their specific topology. A related study employed the J-integral method to analyse the fracture toughness of octet-truss lattices, reporting a linear increase in fracture toughness with relative density and the square root of the cell size. ¹¹⁵ It was also demonstrated that the stress intensity factor, as used in conventional elastic fracture mechanics, is insufficient to characterise fracture in these structures. ¹¹⁶ The traditional testing of fracture protocols is inadequate to characterise such structures. As such, the study developed a general test and design protocol for fracture analysis of octet-truss lattices based on a fracture mechanism map. The map is a plot of normalised toughness as a function of normalised T-stress and relative density. The T-stress is a parameter that captures the effect of crack size and stress triaxiality on fracture toughness. The fracture mechanism map can be used to predict the failure mode (strut tensile fracture or strut elastic buckling) and the toughness of the octet-truss metamaterial for a given set of material and structural parameters. The researchers also noted that the equivalent of plane-strain fracture toughness in three-dimensional metamaterials cannot be accurately assessed with conventional through-thickness cracked samples. They suggested that crack specimens specifically designed to negate surface effects are required for this type of measurement. ¹¹⁶ A study by Wang et al. ¹¹⁷ reported on the fracture behaviour of octet truss and Schwarz P shell 3D lattices at increasing relative densities. The structures were fabricated via Projection micro-stereolithography (PμSL) and the J-integral method based on ASTM E1820 was adopted to calculate the fracture toughness. This research indicated that Schwarz P shell lattices showed greater resistance to crack initiation (fracture initiation toughness) than traditional octet truss lattices, with average improvements reaching 150%. The authors also demonstrated that their polymer octet truss and Schwarz P shell lattices had higher normalised fracture toughness than most existing polymer and metal truss- and plate-based lattices. The exception was polymer octet and kagome truss micro-lattices made from IP-Dip using two-photon lithography (TPL), which featured dimensions as small as 1.03 to 2.25 μm. ¹¹⁸ The high fracture toughness in this article can be attributed to the differences in intrinsic materials and characteristic size scales. The fracture toughness-relative density relationship was described according to the expression in equation 8.

$${\displaystyle \frac{K_{IC}}{{\sigma }_{ys\sqrt{L}}}=D^{{\overline{\rho }}^d}{\left({\displaystyle \frac{{ϵ}_f}{{ϵ}_{0s}}}\right)}^{\frac{n+1}{2n}}}$$

(8)

Figure 32.

The topology of various structures investigated for fracture toughness by Choukir and Singh ¹¹⁴ (Copyright obtained from Elsevier Ltd).

Where: ε_f = fracture strain, ε_0s = yield strain, and σ_ys = yield strength of the constituent material n is the strain hardening exponent. L is the unit cell length. D and d are coefficients for the lattices and are topology-dependent.

Maurizi et al. ¹¹⁸ investigated the fracture behaviour of 3D nano-architected lattices, focusing on the octet and 3D Kagome structures. Nano-architected octet lattices demonstrated a fracture initiation toughness that was eight times higher than previously achieved in macroscopic octet titanium structures. This fracture toughness value was determined through a combination of tensile fracture experiments conducted in situ and finite element simulations. The structures demonstrated comparable or better performance than those reported in the literature, such as 2D brittle lattices, 2D elastoplastic lattice materials, and 3D macro-lattices. It also shows that the fracture toughness of the metamaterials is independent of the thickness of the unit cell members.

The influence of the orientation of the octet-truss lattices on the fracture toughness was investigated by Li, Pavier, and Coules and revealed that lattice orientation has significant effects on the crack paths; the cracks propagated horizontally to the initial crack for X and Z orientations whereas the cracks deviate from the initial at an angle of 30° for Y-oriented structures. ¹¹⁹ A similar study reported the influence of build orientation on the fracture behaviour of 3D-printed PLA octet lattices. ¹²⁰ It was shown that when the printing direction is parallel to the notch plane, mode II fracture (delamination) occurs, followed by ductile failure of struts. This fracture leads to higher fracture toughness. On the other hand, when the notch plane and build direction are perpendicular to each other, it was shown that brittle (mode I) fracture occurs along the notch path, leading to lower fracture toughness. These results are illustrated in Figure 33. The article further demonstrated that the fracture toughness and peak loads for both build orientations increase with the relative densities.

Figure 33.

The influence of 3D printing orientation of PLA octet lattices. (a) The notch plane and build direction are parallel, and (b) the build direction and notch plane are perpendicular to each other. The fracture modes II (delamination and ductile failure) and I (brittle fracture) are shown for parallel and perpendicular build orientations, respectively, for PLA samples printed at 27% and 61% relative density ¹²⁰ (Adapted with copyright permission from Elsevier Ltd).

Studies have reported that disorder in lattice structures enhances the fracture toughness by generating distributed damage. Fulco et al. ¹¹³ recently (in 2025) investigated 2D plane strain triangular lattices where the repeating unit cells were of a size comparable to the effective process zone. The introduction of geometric disorder was incorporated by perturbing the node locations in 2D. The study utilised 2D linear elastic finite element simulations to calculate the fracture behaviours of lattices with varying geometries. Experimental analysis employing photoelasticity facilitated the visual assessment of material degradation leading to structural failure. The data indicated that under conditions of optimally introduced irregularity, the material's resistance to fracture exceeded that of a regularly structured lattice of comparable density by a factor exceeding 2.6. This suggests that the incorporation of disorder within such structural frameworks serves to augment their fracture toughness. Karapiperis and Kochmann ¹²¹ reported on the graph neural networks for the prediction of fracture pathways in irregularly architected materials and revealed that disorderly enhances fracture energy. In another study, Choukir et al. ¹²² (2025), explored the effect of disorder on mechanical properties of Voronoi networks-inspired metamaterials. Their findings demonstrated that the introduction of disorder in network structures led to a 20% improvement in strength and a remarkable 100% enhancement in toughness when compared to regular hexagonal honeycombs. This outcome effectively overcomes the inherent strength-toughness compromise. In a related study, ¹²³ the effect of structural irregularity on the mechanical attributes of a two-dimensional auxetic lattice with a re-entrant hexagonal design was examined. The research demonstrated that the direction of applied forces dictated the failure mechanism, resulting in either brittle or ductile behaviour. Tensile strength under brittle failure conditions was found to be dependent on the level of disorder, a relationship not observed in ductile failure scenarios.

Ryvkin and Shraga ¹²⁴ investigated self-similar hierarchical honeycombs through numerical and experimental methods based on the Representative Cell Method and discrete Fourier transform for fracture toughness. It was reported that for the first-order hierarchical honeycomb, Mode I fracture toughness increased by 2.1% compared to the basic honeycomb, where the Mode II fracture toughness increased by 3.6%. For the Second-order hierarchical honeycomb, Mode I fracture toughness increases by 9.2% over the basic honeycomb, whereas Mode II fracture toughness increases by 20%. These results demonstrate that increasing the hierarchical order enhances fracture toughness, particularly at lower relative densities.

Deformation mechanisms

Fundamentals

Metamaterials undergo different deformations during loading depending on the geometry of the unit cells and the configuration of the unit cells in the metamaterial. ¹²⁵ The behaviour of the specific and interconnected unit cells is critical to evaluating the mechanical deformations of metamaterials. Similar to traditional materials, metamaterials can be developed to undergo elastic deformation, plastic deformation or topology-transition-induced deformation. The deformation mechanisms of metamaterials can be harnessed to design tuneable structures for specific applications, especially where geometry variation would be necessary upon the introduction of external stimuli such as force, heat, electricity and magnetic fields.

Generally, the honeycomb structure experiences various deformation regimes during in-plane compression loading, as earlier described in Figure 22 and further shown in Figure 34. These regimes are (I) onset of yielding, (II) plateau region, (III) onset of densification, and (IV) densification. During regime (I), the structure undergoes elastic deformation and can recover upon withdrawal of the applied load until there is an onset of yielding. In this regime, the structure remains undistorted and can recover to the original shape and size upon release of the applied force (Figure 34). At the plateau region, there is uniform collapse of the structure, i.e., without bulging and shear bands start to form in the middle of the structure. At the end of the plateau region, the structure has fully closed, and the structural members of the honeycomb cells are in contact with each other, leading to densification (see Figure 35). Kucewicz et al. ¹²⁶ presented a failure model for honeycomb cellular structure through numerical and experimental tests. During the compression test, a similar deformation to the idealised one in Figure 35 was observed (see Figures 36 and 37). As the compressive load increases, a shear plane is formed at 45° to the load direction (regime 1), followed by further collapse of the whole structure (regimes 2 and 3). At this stage, the shape of the unit cell becomes distorted to a nearly rectangular shape. Erosion and breakages occur at the joints of the members of the unit cell. The mechanical deformation of a honeycomb structure depends on wall thickness, arrangement order, shape, and Poisson's ratio as reported by Yang et al. ¹²⁷ To study the deformation behaviour of honeycomb cells during compression loading, the structure can be modelled as a homogeneous material, as shown in Figure 38. ¹²⁷ The unit cell is modelled symmetrically as AB and BC members, and the equilibrium condition of the force can be applied to evaluate their deformation (Figure 38).

Figure 34.

Deformation regimes during in-plane compression loading of honeycomb structures ¹²⁸ (adapted through open access under the creative commons attribution (CC BY) license).

Figure 35.

Idealised deformation of Honeycomb core with a hexagonal form. The unit cell transforms from hexagonal morphology to rectangular (plateau region), re-entrant (plateau up to the onset of densification), and interlocking triangles/star-like structures.

Figure 36.

In-plane compression deformation of honeycomb cellular structure (a) force versus displacement and (b) progressive stages of honeycomb structure deformation, at the initial state (elastic region), regions 1, 2 and 3 (plateau regime). ¹²⁶ (reused under the creative Commons CC-BY-NC-ND license from Elsevier Ltd).

Figure 37.

Deformation of the honeycomb unit cells during in-plane compression loading. Both experimental and the FEA for three different modelling methods. ¹²⁶ (Reused under the Creative Commons CC-BY-NC-ND license from Elsevier Ltd).

Figure 38.

Schematic representation of modelling of hexagonal honeycomb unit cell during compression in Y-direction: (a) unit cell under axial compression (Y-direction), (b) stress distribution along ABC segment, (c) loading on segment AB, and (d) segment BC. ¹²⁷ (reused from MDPI under open access through the creative commons attribution (CC BY) license).

Case studies

Deformation of re-entrant honeycomb metamaterials has been reported. For example, a study by Gao et al. ¹²⁹ employed a finite element analysis in ABAQUS/Explicit to compare the compressive and flexural behaviour of re-entrant honeycomb, 3D re-entrant lattice, and regular hexagonal honeycomb (Figure 39). The numerical simulations undertaken at different strains of 0%, 20%, 40%, 60%, and 80% revealed deformation mechanisms (see Figure 40). For the hexagonal honeycomb structure, the shear planes appear perpendicular to the loading direction, and the top and lower layers of the structure experience the minimum stresses. For re-entrant honeycomb, the top layer of the unit cells experiences maximum stresses upon full loading. On a 3D re-entrant lattice, there is continuous deformation of the structure and a uniform stress distribution is observed. ¹²⁹ Zhang et al. ¹³⁰ reported several findings regarding the deformation of 3D re-entrant and hexagonal meta-materials. It was observed that the 3D re-entrant honeycomb can withstand substantial bending without incurring damage. Under a bending load, the 3D hexagonal honeycomb showed an inverted trapezoidal mode of deformation at its mid-span, in contrast to the 3D re-entrant honeycomb, which deformed into a trapezoidal shape in its central region. The hexagonal honeycomb's mid-span section takes on an inverted trapezoidal shape due to the upper surface being squeezed and expanded out-of-plane, and the lower surface being stretched and shrunk. In contrast, the re-entrant honeycomb's mid-span section assumes a trapezoidal shape, a result of its auxetic effect, where the upper surface shrinks under compression and the lower surface expands under tension.

Figure 39.

Compressive simulations of honeycomb structures (a) hexagonal, (b) re-entrant, and (c) 3D re-entrant. ¹²⁹ (reused from MDPI under open access through the creative commons attribution (CC BY) license).

Figure 40.

Simulations of compression loading of the structures (a) validation of simulations (b) hexagonal, re-entrant, and 3D re-entrant honeycomb. ¹²⁹ (reused from MDPI under open access through the creative commons attribution (CC BY) license).

The study reported further that the deformation of the re-entrant honeycomb unit cells can be divided into four stages. In stage II, the unit cell undergoes expansion in a perpendicular direction to the tensile stress due to the auxetic effect. This is followed by regular deformation in which the unit cell no longer has the auxetic effect (stage III). The deformation continues until the collapse of the unit cell in stage IV (see Figure 41). In the figure, stages II and III are defined as transition stages, in which the re-entrant unit cell exhibits the auxetic effect. The hexagonal unit cell does not undergo the transition stage deformation and hence does not have auxetic characteristics. A study by Lian et al. ¹³¹ demonstrated the deformation behaviour of re-entrant honeycomb unit cells in a hierarchical metamaterial structure, as shown in Figure 42. Under compressive loading, the stress-strain curves indicated that the re-entrant hierarchical auxetic (RHA) structure exhibits a distinct shock zone before entering the plateau region. However, in contrast to the conventional honeycomb structure, the RHA demonstrates two separate plateau periods (Figure 42(a)). It was also shown that the honeycomb's initial deformation (first-stage collapse) involves the simultaneous rotation and compression of its side walls at the macroscopic level. At this point, the substructure of the honeycomb's horizontal and inclined edges remains largely unchanged (Figure 42(b). During the second-stage collapse, the honeycomb's side undergoes compression and becomes compacted, with further deformation limited to the bottom side (Figure 42(c)). As shown in Figure 42(d), the substructure transforms during compression, changing from what resembles a diamond to an approximate square, and the dominant deformation is plastic rotation at the nodes. ¹³¹

Figure 41.

Deformation stages of re-entrant unit cell during a tensile deformation as presented by ref. ¹³⁰ The hexagonal unit cell does not involve the transitional stage in which the re-entrant structure exhibits auxetic effect. (Copyright obtained from Elsevier Ltd).

Figure 42.

Illustrating the deformation mechanisms of hierarchical re-entrant honeycomb auxetic structure: (a) stress-strain curves during compression loading showing the two plateau regions, (b) deformation during the first stage collapse, (c) deformation during the second collapse, and (d) morphology transformation during the second stage collapse. ¹³¹ (reproduced from scientific reports under a creative commons attribution 4.0 international license).

Table 5 summarises some of the reported deformation mechanisms of common mechanical metamaterials. It can be deduced that during deformation evaluations, the behaviours of the unit cell, cell linkages, and whole structure (metamaterial) are usually considered.

Table 5.

Summary of the failure mechanisms of the common mechanical metamaterials reported in the literature.

Structure	Loading conditions	Failure mechanisms
3D Double-arrowhead (DAH) auxetic metamaterials 132	Quasi-static uniaxial compression	Based on the loading direction and relative density: (1) Layer-by-layer collapse followed by global instability. The short and long struts are gradually folded. (2) Initially, the shorter struts fractured due to tensile forces, which was then followed by a sequential, layer-by-layer breakdown of the structure. (3) Plastic hinge formation, which is similar to BCC structures during compression.
Chiral metamaterials based on Euclidean tessellations (Chiral honeycombs) 133	Uniaxial compressive loading tests (high-strain behaviour conditions)	Layer-by-layer collapse of the non-adjacent ligaments of the structure.
Four-point star-shaped structures 52	Uniaxial compressive loading tests	(1) Shape distortion of the unit cells (2) Bending of the intercell linkages (3) Crack initiation at the intercell links at the joints with the unit cells (4) Crack growth and propagation, leading to breakages of the links (5) Catastrophic failure of the unit cells and detachment of the intercell links.
Four-Star-Honeycomb Hybrid Metamaterial 134	In-Plane Compression loading	1) Shape distortion of the unit cells 2) Bending of the intercell links 3) Cracking and fracture of the struts of the unit cells and interlinks. 4) Collapse of the unit cells and the metamaterial 5) Bulging and buckling of the metamaterial
Re-entrant hexagonal structure 135	Uniaxial tensile loading	Realignment of the cell hinges within the re-entrant structure. Transformation of the re-entrant structure into a rectangular geometry.
C-entrant (CRM), V-entrant (VRM), and S-entrant (SRM) metamaterials 136	Uniaxial compression test	For VRM and CRM, deformation starts at the centre of the structure, forming an “X” shear band. This is followed by rotation and deformation of the ligaments of the re-entrant. At high strains, the deformation occurs in one direction and layer-by-layer collapse is observed. For SRM, uniform deformation occurs at low and high strains and results in full compression.
Re-entrant circular auxetic honeycombs 137	Compression loading	X mode of deformation at the initial stages of loading (the shear planes cross each other as X). An increase in loading leads to the top cell walls moving towards the bottom ones. This eventually leads to the crushing of the structure of the unit cells.
Origami-based metamaterials (Tachi–Miura polyhedra (TMP)) 138	Quasi-static bending loading	The deformation mechanism changes from bending to auxetic compression. During the global bending of the structure, the unit cells do not distort/shrink. The auxetic compression involves the shrinking of the TMP unit cells.
3D chiral metamaterials 139	Compression and uniaxial tensile loading	(1) Spatial rotation of chiral nodes (2) Bending of the ligaments (3) Tensile shearing (4) Compression shrinkage
Anisotropic tetra-chiral metamaterials (with elliptic nodes) 140	Tensile loading	Rotation of the nodes, stretching of ligaments.
Rotating squares and anti-tetrarchical systems 141	Tensile loading	Rotating Square System: When a uniaxial load is applied to the rotating square system, the square units rotate. Anti-Tetrachiral Honeycomb System: When the uniaxial load is applied to the anti-tetrachiral honeycomb system, the square units rotate, and the ligaments flex

Design strategies for mechanical metamaterials

Conventional design

In designing metamaterials, several factors are considered to achieve the desired performance. The functional requirement of a metamaterial dictates the structure's desirable properties and overall design. Based on existing literature, a framework demonstrating a strategy for designing metamaterials is presented in Figure 43. The desirable properties determine the design of the unit cell, which entails structure, topology and density optimisation. The unit cell design and assembly are then evaluated for performance and mapped against the desirable properties. This is usually undertaken through computational modelling and rapid prototyping, followed by experimental design validation. The metamaterial design involves: (1) unit cell design and (2) configuration of the unit cell assembly to achieve the desired properties. The first strategy in metamaterial design heavily relies on the designer's experience, employing a trial-and-error method followed by evaluating mechanical properties and responses, which involves experimental mechanical testing or computer simulations. This is the most used strategy; designers rely on natural or artificial inspiration. In this way, metamaterials inspired by biology,^142,143 scissor-configuration ¹⁴⁴ and other structures have been designed. In a recent commentary, it was highlighted that state-of-the-art computational design approaches are desired to develop several art-inspired metamaterials. ¹⁴⁵ For example, to achieve desired physical properties with complex shapes in origami design (geometry of folds), specific mathematical formulations were presented to govern the formation of folds. ¹⁴⁵ A design approach for two-dimensional multi-stable cellular metamaterials was developed, utilising local rotational degrees of freedom to enhance energy absorption under various mechanical loads by Pechac and Frazier. ¹⁴⁶ By precisely aligning the rotation centres of two layers with identical but opposite chirality, the approach led to a design that enables local snap-through events during in-plane deformation. These events significantly contribute to global mechanical energy absorption. Furthermore, the rotational symmetry of the (quasi-)crystalline layers determines the directional energy absorption characteristics. ¹⁴⁶

Figure 43.

A conceptual framework demonstrating key factors for consideration during the design of metamaterials.

Topology optimisation

The second strategy for metamaterial design is topology optimisation. The objective of topology optimisation is to determine the optimal material distribution within a geometry using specific materials based on governing equations, as well as considering a specified set of loads, boundary conditions, and constraints. ¹⁴⁷ Generally, the formulation of a topology optimisation problem involves defining several key elements that guide the optimisation process:

Objective Function: This function represents the quantity that is to be minimised or maximised to achieve the best performance. Common objectives include minimising compliance (to maximise stiffness), minimising stress, or minimising overall material volume. The ability to incorporate multiple objectives is also gaining prominence. ¹⁴⁸

Design Variables: These variables describe the distribution of material within the designated design space. In discrete formulations, material is either present (represented by a value of 1) or absent (represented by 0). In continuous formulations, the material density can vary continuously between 0 and 1 at each location.

Design Space: This defines the allowable volume or region within which the design can exist. Considerations such as assembly requirements, packaging constraints, and accessibility for tools or human interaction are factored into identifying this space. Specific regions or components that cannot be modified during the optimisation process are designated as non-design regions.

Constraints: These are characteristics that the final solution must satisfy. Common examples include a maximum allowable amount of material to be distributed (volume constraint) or limits on maximum stress values within the structure.

A key development in modern topology optimisation (TO) for metamaterials is the growing emphasis on manufacturing constraints. In additive manufacturing, these include overhang and member size limits, which help reduce support structures, material waste, and overall production costs. ¹⁴⁹ The TO approach holistically balances performance goals, material efficiency, and practical manufacturability. For metamaterials with complex architectures, incorporating such constraints ensures designs are both theoretically optimal and physically manufacturable.

Topology optimisation (TO) differs fundamentally from sizing and shape optimisation by offering far greater design freedom. While sizing optimisation adjusts parameters like thickness or cross-sectional area, and shape optimisation alters existing boundaries, TO enables the creation of entirely new structural configurations. ¹⁵⁰ It operates independently of an initial design and allows for complex changes such as the introduction of new holes, which is vital for designing advanced metamaterials. This freedom enables the discovery of innovative, high-performing microstructures that are often beyond the reach of conventional design methods.

Topology optimisation employs various methods to achieve optimal material distribution, each with distinct strengths and limitations. The most widely used approaches include density-based methods, level-set methods, and evolutionary structural optimisation (ESO/BESO). ¹⁵¹ These techniques are especially relevant in the design of complex metamaterials, where precision and adaptability are crucial. However, each method poses its challenges when addressing the intricate geometries and performance demands of such structures. These techniques are briefly described as follows:

Density-Based Methods (SIMP)

Density-based methods are a commonly used approach in topology optimisation. They work by dividing the design space into discrete elements and assigning a material density to each one. This density is treated as a continuous variable, allowing gradual material distribution across the structure. ¹⁵² The Solid Isotropic Material with Penalisation (SIMP) method is one of the most widely used interpolation techniques in density-based topology optimisation. It models material properties, such as Young's modulus, as a power-law function of an element's pseudo-density. A penalisation factor, typically between 2 and 4, is applied to discourage intermediate densities and promote a clear binary material distribution. To maintain meaningful gradients in the optimisation process, a small lower bound is set on Young's modulus, even when density values approach zero. SIMP effectively converts a discrete optimisation problem into a continuous, nonlinear programming problem over a convex design space.

In gradient-based optimisation, accurate sensitivity analysis is essential. Within the SIMP framework, this involves computing the derivative of the objective function, such as structural compliance, with respect to the element densities. ¹⁵³ The continuous adjoint method is commonly used for this, enabling efficient sensitivity calculations across many design variables. Additionally, the concept of a topological gradient helps interpret SIMP and informs the selection of the penalisation function.

Despite its popularity, SIMP has several limitations. It can yield intermediate or “grey” densities that are hard to interpret physically and difficult to manufacture. The method's performance is sensitive to the penalisation factor and may result in numerical issues such as spurious holes. While it offers computational efficiency and ease of implementation, extra steps such as filtering and projection are often required to refine the design. For metamaterials, where sharp material boundaries are crucial, these grey areas pose a significant challenge, underlining the need for more robust regularisation techniques.

• (b)

Level-Set Methods

Level-set methods provide a powerful alternative for topology optimisation by representing geometric boundaries implicitly. The geometry is defined as the zero-level set of a higher-dimensional function, offering a clear and precise description. In this method, regions with negative values represent material, zero denotes the boundary, and positive values indicate voids. ¹⁵³ The boundary evolves dynamically according to the Hamilton-Jacobi equation, guided by a normal velocity field. This approach enables smooth and accurate tracking of complex design changes over time.

In level-set methods, sensitivity analysis often relies on the concept of a topological derivative, which measures how a cost function changes when a small hole or inclusion is introduced into the structure. This information guides where material should be added or removed to improve the design. ¹⁵⁴ The resulting sensitivity is used as the normal velocity in the Hamilton-Jacobi equation, controlling the evolution of the level-set function. The adjoint variable method is commonly employed for this purpose due to its computational efficiency. A key strength of level-set methods is their natural ability to manage complex topological changes, such as merging, splitting, or the emergence of new boundaries, without requiring re-meshing. This is particularly advantageous for metamaterial design, where intricate and evolving geometries are essential for achieving specific properties. Their precision and flexibility make them especially well-suited for generating clear and functionally effective microstructures, despite their higher computational demands.

However, level-set methods tend to be more computationally intensive and less efficient than density-based approaches. They may also be sensitive to the initial shape of the level-set function. Additionally, some versions struggle to introduce new material interfaces unless the topological derivative is explicitly incorporated.

• (c)

Evolutionary Structural Optimisation (ESO/BESO)

Evolutionary Structural Optimisation (ESO) and its advanced variant, Bi-directional ESO (BESO), are widely used and intuitive topology optimisation methods. They work by gradually modifying material distribution; ESO through removal and BESO through both removal and addition, to improve structural performance. ¹⁵⁵ While easy to implement and understand, they may offer less precision than gradient-based approaches.

The core principle of ESO is to iteratively remove inefficient material from an initial design to achieve maximum stiffness and minimum weight. The process begins with defining the design domain, typically discretised via Finite Element Analysis, followed by applying loads and calculating structural responses such as stress distribution. ¹⁵⁶ Material under low stress, often identified using the von Mises criterion, is removed in successive cycles until an optimal layout or volume target is reached. BESO enhances this method by allowing both removal and addition of material during each iteration, addressing ESO's unidirectional limitation.

ESO is valued for its simplicity, ease of implementation with FEA tools, and low computational cost, making it an attractive option for early-stage design. ¹⁵⁶ Although it lacks a fully developed theoretical foundation, its results often align well with analytical solutions, and recent research has begun linking it to more rigorous methods such as SIMP and linear programming. Limitations include potential convergence issues, sensitivity to local optima, and dependence on the material removal strategy used. Despite these drawbacks, its practicality and scalability make it useful, particularly for large-scale or conceptual design tasks. For metamaterial design, ESO/BESO offers an accessible starting point, even if more advanced methods are needed for final optimisation. ¹⁵⁷

Table 6 provides a clear, organised comparison of the main topology optimisation methods based on published literature. It outlines their core principles, representation of design variables and geometry, ability to handle topological changes, computational requirements, and key strengths and limitations. Such an overview will be suitable for choosing the most suitable TO method based on metamaterial design goals, computational capacity, and manufacturing constraints. By simplifying complex technical details, it supports informed and efficient decision-making in the design process.

Table 6.

Comparison of key topology optimisation methods.

Method	Principle	Design Variable	Geometry Representation	Handling of Topological Changes	Computational Cost	Advantages	Limitations
SIMP	Penalises intermediate densities to achieve binary material distribution.	Continuous density (0–1).	Element-wise density.	Indirectly, through density changes.	Moderate to High.	Computationally efficient, easy to implement.	Produces intermediate densities (“grey areas”), sensitive to penalty factor, may have spurious holes.
Level-Set	Implicitly represents geometry as an iso-contour of a level-set function.	Parameters of the level-set function.	Crisp boundary representation.	Naturally handles merging and splitting.	High.	Clear boundaries, handles complex geometries and topological changes naturally.	High computational cost, sensitive to the initial function.
ESO/BESO	Iterative removal (ESO) or addition/removal (BESO) of inefficient material.	Discrete elements (present/absent).	Explicit element removal/addition.	Direct, through element removal/addition.	Relatively Low.	Simple to program, less dependent on initial design, applicable to large-scale problems.	Poor convergence, susceptible to local optima, lacks a solid theoretical basis (historically).

BESO, bi-directional evolutionary structural optimisation.

Topology optimisation is effectively employed to design both 2D and 3D mechanical metamaterials exhibiting auxetic and anti-auxetic behaviours. ¹⁵⁸ The design objectives for these materials often include maximising the negative Poisson's ratio, enhancing load-bearing capacity, and reducing stress concentrations. Computational frameworks for such designs typically combine density-based topology optimisation with nonlinear homogenisation methods, enabling optimisation across large strain ranges. This approach also extends to multi-material configurations, allowing for the creation of complex structures with tailored mechanical properties. Common examples of optimised microstructures include re-entrant and chiral geometries, which are well known for their auxetic characteristics. This application underscores the power of topology optimisation in engineering extreme mechanical responses through microstructural design. By precisely tailoring the internal architecture, TO enables the development of materials with unconventional behaviours, opening new possibilities in applications requiring bespoke deformation, energy absorption, or stiffness profiles.

Topology optimisation is employed to design composites that incorporate negative-stiffness inclusions (structures which deform in a direction opposite to the applied force), which are stabilised by a surrounding matrix material. ¹⁵⁹ This stabilisation mechanism can lead to extremely effective dynamic responses, including significantly enhanced damping. Such features are particularly beneficial in applications involving vibration isolation and control. Topology optimisation thus provides a systematic framework for harnessing otherwise unstable material responses in functional, manufacturable designs. Beyond auxetic and negative stiffness behaviours, topology optimisation is also applied to the design of metamaterials with extreme thermal expansion, extreme bulk and shear moduli, desirable bandgaps for wave attenuation, enhanced damping, and high energy absorption capabilities. ¹⁵⁹ These tailored mechanical responses are made possible by controlling microstructure at a fine scale through optimisation algorithms. TO offers versatility in meeting multifunctional performance criteria within a single material system. As the complexity of design objectives increases, advanced TO methods enable fine-tuning of physical responses across a range of engineering applications. This makes TO a powerful design tool in the development of next-generation mechanical metamaterials.

Several studies have used topology optimisation methods in the design of mechanical metamaterials. ¹⁶⁰ For example, using an algorithm based on the bi-directional evolutionary topology optimisation and energy-based homogenisation approach, Chatterjee et al. ¹⁶¹ explored the impact of material uncertainty on the optimal design of mechanical metamaterials. It was also demonstrated that robust design improves the optimal mean performance of metamaterials and elastic properties, thereby making the design less sensitive. Another study ¹⁶² utilised finite element-based topology optimisation to design soft hyperelastic biomaterials that maximise lateral expansion and mimic the specific directional properties of human skin. The study employed a solid isotropic material with penalisation (SIMP)-based topology optimisation method, utilising Abaqus software with the Tosca module to iteratively adjust material density. Key findings revealed that topology-optimised designs exhibited significantly enhanced auxetic properties, with lateral expansion increased by over 50% and lower maximum stress compared to initial patterns; for instance, optimised I-shape and slit designs showed the greatest expansion. Furthermore, the re-entrant shape demonstrated the ability to replicate the anisotropic stress and elastic modulus of human skin, whilst optimisation transformed non-auxetic triangular shapes into auxetic ones. The authors concluded that this pioneering application of topology optimisation to soft hyperelastic biomaterials successfully created designs with enhanced expansion and tailored anisotropic behaviours, promising for biomedical applications requiring accurate representation of skin deformation. A study by Chen and Huang ¹⁶³ developed a systematic topology optimisation approach for designing 3D chiral metamaterials based on couple-stress homogenisation, which allows for the calculation of effective moduli of a 3D material through a series of numerical tests on its unit cell. A key finding in this work was that by optimising specific components of the coupling matrix, various chiral metamaterials with distinct topological patterns can be obtained. The results further demonstrated that the chirality of the designed metamaterials is evident in their twist deformation under uniaxial compression, and the coupling and curvature matrices are size-dependent. The proposed topology optimisation method effectively designs 3D chiral metamaterials by characterising their inherent chirality through couple-stress homogenisation and maximising the coupling effects between stress and curvature. Another study evaluated Topology Optimisation (TO), Lattice Structures (LS), and Lattice-infilled Topology Optimisation (LTO) for Ti-6Al-4V in complex geometries, employing a novel FBCore hybrid metamaterial unit cell that merged FCC and BCC designs. ¹⁶⁴ It was indicated that FBCore significantly outperformed other designs, with TO achieving 61.1% mass reduction, LS achieving 78.1%, and LTO offering a balanced 67.8% reduction while maintaining structural integrity. The study concluded that LTO provided optimal performance and material efficiency by combining the benefits of TO and LS, validated with an average error of just 3.5%.

Zhang et al. ¹⁶⁵ utilised topology optimisation to design mechanical metamaterials with the objectives of achieving maximum stiffness and minimum Poisson's ratio, and the optimisation achieved optimal stiffness distribution of the design area. The topology optimisation algorithms can be classified into two categories, i.e., gradient and intelligent algorithms. Techniques such as the solid isotropic material with penalisation method, the level set method, the phase field method, and moving morphable components are examples of gradient-based algorithms, whereas genetic and machine learning (ML) algorithms are examples of intelligent algorithms. ¹⁶⁶ The gradient algorithms are traditional forward design processes, whereas intelligent algorithms are known as inverse design strategies. As shown in Figure 44(a), the forward design involves choosing a design from nature/artificial inspiration, assigning its materials, and evaluating load-displacement properties. This inverse design approach begins with an optimised model, the output of a gradient algorithm, as the starting design population to match the desired curve. Once the population's gene data has been processed, it is converted into a geometric model for subsequent numerical analysis (Figure 44(b)).

Figure 44.

Topology optimisation strategies for metamaterials (a) traditional/forward and (b) inverse design process ¹⁶⁶ (reused freely as open access under creative commons CC BY license).

The implementation of topology optimisation for designing metamaterials depends significantly on advanced computational tools and software packages. These tools range from robust commercial suites, such as COMSOL and ANSYS, to highly customisable open-source platforms like MATLAB-based codes and FEniCS. Commercial packages offer user-friendly interfaces and integrated solvers, facilitating rapid prototyping and design iteration. In contrast, open-source tools provide greater flexibility for research-driven customisation and the development of novel algorithms. Herein, we summarise common tools for the benefit of our readers in their research (Table 7).

Table 7.

Overview of commercial and open-source TO software for metamaterials.

Software Name	Type	Key Features	Metamaterial Relevance	Strengths	Limitations
Altair OptiStruct/Inspire	Commercial	Generative design, TO, shape optimisation, stress/displacement constraints, gravity/temperature loads, manufacturing constraints (symmetry, draw direction, overhang).	General structural TO, can be applied to unit cell design for mechanical metamaterials. 167	Industry-leading, robust, integrated with full CAE suite, strong AM support.	General-purpose, may require expert setup for highly specialised metamaterial physics. 167
nTop (nTopology)	Commercial	Automated geometry reconstruction, design for AM, integrated simulation, latticing, ribbing, volumetric representation. 168	Explicitly supports “meta-materials” (pre-programmed lattices) for weight savings and mechanical performance. 169	Excellent for AM, complex geometry handling, and user-friendly workflows.	Targets expert users for customisation (Element Pro). 169
ParaMatters CogniCAD	Commercial	Cloud-based, mechanical performance-driven feature generation, advanced TO, vibrational characteristics, meta-materials, multi-material option (beta). 170	Explicitly mentions “meta-materials” and “mesostructures similar to those found in human bone”.	Streamlined workflow for AM, broad optimisation feature set.	Cloud-based; may have data privacy concerns for some users.
Autodesk Solutions	Commercial	Advanced design/simulation, cloud-based 3D CAD/CAM/CAE, CFD, FEA.	General TO capabilities applicable to metamaterial unit cells. 171	Comprehensive suite, widely used in industry.	Not specifically tailored for metamaterials, may require custom scripting for advanced physics.
COMSOL Multiphysics 172	Commercial	Multi-physics simulation, FEA for EM, acoustic, mechanical, thermal problems.	Used for solving Maxwell's equations in EM TO, acoustic-structure interaction.	Strong multi-physics coupling, flexible for custom physics formulations.	Primarily a solver, TO capabilities may require more manual setup compared to dedicated TO software.
OpenPISCO 173	Open-Source	Modular, Python-based interfaces with CodeAster/FreeFem++, GUI/CLI applications, and implicit modelling.	Research-oriented, flexible for custom metamaterial TO algorithms and material models.	Highly customizable, extensible, good for R&D, no licensing cost.	Requires programming knowledge, community support, not as user-friendly as commercial GUI tools
PLATO Engine 174	Open-Source	Core TO functionality, GPU-enabled solver, generates print-ready designs.	Generates unique conformal lattices for lightweight metamaterials.	Optimised for AM, robust engineering codes from Sandia.	Core functionality open-source, full product licensed for government use.

Machine learning (Ml)

Machine learning (ML) methods have been adopted widely in the design of metamaterials. ¹⁷⁵ The research on ML has been on the increase for the past years according to the Web of Science (WoS) database (Figure 45). In terms of applications, ML enhances the accuracy and efficiency of material characterisation and structural design, as well as fostering computational efficiency in discovering and designing complex structures. Furthermore, it enables the automation of discovery and design processes, facilitating the precise tailoring of mechanical properties from both material and structural perspectives, and resolving dilemmas in response prediction and inverse design. ¹⁷⁶ Machine Learning (ML) significantly accelerates both the design phase and the assessment of material functionality. Once trained, ML models drastically enhance computational speed, performing complex calculations far quicker than traditional methods like Finite Element Analysis, enabling the analysis of vast microstructures in minutes rather than days. Furthermore, ML technologies are crucial for identifying previously unseen trends and complex relationships within design variables, allowing for efficient exploration of the extensive design space to uncover novel and optimal structures. This synergistic integration also aids in solving intricate physical problems with multi-objective requirements and ensures better utilisation of data, even limited datasets, through generative models. Specifically, ML offers the following attributes in metamaterial design.

Surrogate modelling: In this case, Artificial Neural Networks (ANNs) are used to create models that predict material properties, effectively sidestepping the need for time-consuming numerical simulations such as Finite Element Analysis (FEA). This capability is crucial for rapid iteration in the design process.

Inverse design: It allows algorithms to map the desired performance specifications to the corresponding metamaterial structures (design parameters). This reverses the traditional trial-and-error approach, thereby reducing the design time and effort. Furthermore, ML is instrumental in hyperparameter optimisation, where AI fine-tunes the settings and architecture of ANN models to ensure optimal performance and output quality.

Topology optimisation integrated with machine learning (ML) techniques is employed to intelligently distribute material within a design space, maximising performance for microscale elastic structures. ¹⁷⁷ Dealing with vast amounts of data is made easier through dimensionality reduction, where ML methods either generate more compact features or identify the most critical existing ones. Techniques such as ShapeDNA and Principal Component Analysis (PCA) help manage high-dimensional data, reducing computational costs in simulations, predictions, and the generation of new metamaterial designs.

Nature-inspired AI algorithms, including Genetic Algorithms (GA) and various swarm intelligence methods like Particle Swarm Optimisation (PSO) and Ant Colony Optimisation, draw upon natural processes to tackle complex optimisation and design challenges. ¹⁷⁸ These algorithms are exceptionally effective at navigating vast design possibilities and unearthing innovative structures with desirable properties, further pushing the boundaries of metamaterial capabilities.

Figure 45.

A graph of published literature on machine learning (ML) and metamaterials according to the Web of Science (WoS) between 2020 and 2025. There has been growth in the number of publications for the past 5 years.

The machine learning applications in metamaterials design can be broadly categorised into forward and inverse designs. These categories are briefly described below.

Forward Design

Forward design in metamaterials involves predicting the functional properties or performance characteristics of a material given its specific structural parameters.

Multilayer Perceptron (MLP): MLPs are fundamental neural network architectures composed of multiple fully connected layers, which are highly effective for handling complex, nonlinear problems. They are widely utilised for the forward prediction of artificial metasurface performance. For example, MPL was used to approximate the highly nonlinear edge update function within the Metamaterial Graph Network (MGN) framework for mechanical metamaterials analyses. ¹⁷⁹

Convolutional Neural Network (CNN): CNNs are particularly adept at recognising hierarchical spatial structures through their convolutional layers, making them ideal for processing image data and highly effective for forward prediction tasks. For example, CNNs were employed to predict the stress-strain curve of a semi-auxetic mechanical metamaterial. ¹⁸⁰

Large Language Models (LLMs): Emerging research is exploring the capabilities of LLMs (such as ChatGPT, Gemini, LlaMa, and Claude) in the design of metamaterials and material science. ¹⁸¹ For example, text prompt engineering was employed in the pioneering work by Zheng et al. ¹⁸² on generating 3D material microstructures without requiring additional optimisation.

• (b)

   Inverse Design

Inverse design focuses on generating the structural parameters of metamaterials based on desired functionality or performance (such as high strength, negative Poisson's ratio, etc.). This is a more challenging problem than forward design, as a single desired property can often correspond to multiple possible structural configurations, a phenomenon known as the “one-to-many” mapping problem. Inverse design methods can be broadly categorised into three main types:

Exploration-Based Inverse Design: This method focuses on discovering unknown targets, particularly useful when data is scarce and it includes the following techniques:

Reinforcement Learning (RL): Unlike supervised learning, RL does not rely on large, labelled datasets. Instead, an agent learns through interaction with an environment, adjusting parameters and receiving rewards to optimise design.

Monte Carlo Tree Search (MCTS): This advanced tree search algorithm leverages random simulations and exhibits strong heuristic search capabilities. It is frequently integrated with reinforcement learning (RL) as a strategy optimisation technique, enabling the identification of potentially optimal paths and solutions even in the absence of complete data.

Particle Swarm Optimisation (PSO): PSO maximises objective function through guiding a swarm of particles within the search space by adopting heuristic and intelligent exploratory strategies to perform global searches in complex and ill-defined design spaces. PSO is capable of easily finding best input parameters in high-dimensional spaces even with limited data by employing cooperation and information sharing between particles. ¹⁸³

Model-based inverse Design: The method uses models based on data or physics to establish the relationship between the performance and parameters of the metamaterials. The strategies adopted for this method are discussed as follow:

Combining Forward and Inverse Models: This method addresses overfitting through bidirectional exploration between composition and performance. The Machine Learning Design System (MLDS) of copper-aluminium alloys, for example, uses two neural networks: one for prediction of properties from composition (C2P), and another for prediction of composition from properties (P2C). ¹⁸⁴ The P2C model generates candidates, which are then fine-tuned by the more accurate C2P model.

Generative Models: These deep learning models generate new data points, directly creating material compositions or structures based on target performance. Techniques include Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). Xiang et al. ¹⁸⁵ presented a novel GAN-based machine learning model designed to predict the full-field mechanical behaviour of architected metamaterials. By employing a stepwise mapping strategy, this model accurately forecasted deformation and stress fields across multiple loading stages, addressing limitations found in traditional prediction methods. A study by Wang et al. ¹⁸⁶ used a VAE to map complex microstructures of metamaterials into a low-dimensional, continuous, and organised latent space. This latent space then provided a distance metric for measuring shape similarity, enabling interpolation between microstructures, and encoding meaningful patterns of variation in geometries and properties.

Surrogate Models: These are simplified representations of complex functions that approximate the mapping between design parameters and performance. They enable efficient design optimization and experimental validation by using optimization algorithms to adjust input parameters. Danesh et al. ¹⁸⁷ developed surrogate models for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids. The capability of surrogate modelling facilitates swift inverse design and rapid exploration of the design space, addressing the computational expense of traditional simulation methods.

Optimisation-Based Inverse Design: This method presents inverse design as an optimisation problem, tuning material composition/structure through optimisation algorithms to provide a desired target performance/behaviour. Some of the techniques include the following.

Bayesian Optimisation: This combines Bayesian inference and Gaussian processes to dynamically adjust design parameters, improving optimisation efficiency and accuracy. For example, Kuszczak et al. ¹⁸⁸ introduced a computational framework leveraging Bayesian optimisation for the design of hexagonal honeycomb metamaterials with enhanced mechanical properties through varying strut thickness. The study successfully identified optimised configurations that significantly increased both elastic stiffness and plastic or buckling strength, thereby demonstrating the efficacy of Bayesian optimisation and solid material redistribution in improving mechanical metamaterial performance.

Genetic Algorithms (GAs): GAs simulate natural selection to search for optimal configurations, handling complex molecular designs with constraints and multi-stage optimisation. Several studies on the application of GA in the design of metamaterials are available in the published literature. ¹⁸⁹ Particularly, these algorithms are effective in exploring vast, high-dimensional design spaces, which helps in identifying novel and optimal configurations that are otherwise impossible to achieve with traditional techniques. ¹⁹⁰

Several studies are available on the applications of ML in the design of mechanical metamaterials. As per these studies, ML is used in discovery, optimisation, and predictive modelling of metamaterials. A recent review by Zheng et al. ¹⁹¹ titled “Deep Learning in Mechanical Metamaterials: From Prediction and Generation to Inverse Design,” detailed the potential of deep learning in predicting properties, generating geometries, and the inverse design of mechanical metamaterials. It also emphasised the possibility of using deep learning to create universally applicable data, intelligently designed metamaterials and material intelligence. However, deep learning methods are data-intensive, and as such, other ML techniques such as random-forest-based interpretable generative inverse design (RIGID) have been used to design metamaterials. ¹⁹² Zhou et al. ¹⁹³ designed and optimised re-entrant anti-tetrachiral (RATC) structures using a combination of deep learning and genetic algorithms. The study explored the relationship between the design parameters and mechanical response of these structures, specifically stiffness, weight, and Poisson's ratio. These parameters include the radius (r [0,6]), dimensionless geometric parameters ( ${\alpha }_x$ [0,12], ${\alpha }_y$ [0,12], $\beta$ [0,1], $\gamma$ [0,1], $\theta$ [0,90]), the modulus of the material ( $E_0$ [100,10000]), and the Poisson's ratio of the material ( $P_0$ [0, 0.5]). The optimised designs demonstrated significant improvements over the reference structure, including a + 1324% increase in stiffness, a + 187.8% increase in Poisson's ratio, and a + 70.8% reduction in weight. In a related study, Ha et al. ¹⁸⁹ designed, optimised and predicted architected cellular metamaterials using generative machine learning. The cell type, characteristic angle, and the radius-to-length ratio of the inclined strut were used as the design parameters. The research demonstrated a machine learning-driven rapid inverse design methodology, combined with desktop 3D printing, to produce metamaterial designs that match desirable mechanical behaviours. In another study, ¹⁹⁴ the Random Forest (RF) algorithm and the Aquila Optimiser (AO) were employed to enhance the efficiency of metamaterial design and characterisation. The RF algorithm was used for regression analysis to accurately predict Young's modulus, thermal conductivity, and Poisson's ratio, based on various geometric input parameters of metamaterial design. The AO, on the other hand, was used for inverse design optimisation to determine the optimal structural parameters for the desired material properties, such as high stiffness and low thermal conductivity. It was shown that the integration of RF and AO facilitates both property prediction and automated discovery of high-performance metamaterial configurations. ¹⁹⁴

A novel machine learning-assisted model based on the Genetic Programming (GP) algorithm was developed and integrated with theoretical formulations, including a physics-embedded micromechanics model informed by molecular dynamics simulation data. ¹⁹⁵ The model accurately predicted the vibration behaviours of functionally graded graphene origami-enabled auxetic metamaterial (FG-GOEAM) beams immersed in Newtonian fluids. In another study, ¹⁸⁹ a generative machine learning approach was used, specifically utilising generative inverse and surrogate forward neural network models for the rapid inverse design of 3D cubic symmetric, strut-based architectural unit cells metamaterials. The methodology enabled the creation of tailored mechanical behaviours based on user-defined stress-strain curves. A back-propagation framework for lattice-based mechanical neural networks (MNNs) was employed to achieve prescribed static and dynamic performance in intelligent mechanical metamaterials, enabling precise morphing control and on-demand behaviours in a study by Chen et al. ¹⁹⁶

In a study addressing the challenge of mechanical vibration isolation with adaptable payloads, Song et al. ¹⁹⁷ proposed a novel mechanical metamaterial. Their approach involved utilising a genetic algorithm (GA) combined with machine learning (ML) to control the working frequency bandwidth and adjust isolation performance precisely. This facilitated the design of metamaterials capable of achieving quasi-zero stiffness under varied loads, thus enhancing multi-payload adaptability. The designed metamaterial successfully demonstrated accurate control over the working frequency bandwidth and self-adaptive vibration isolation across different payloads. Their work signifies that the Genetic Algorithm-Machine Learning (GA-ML) techniques are crucial for precisely controlling the working frequency bandwidth and adaptively adjusting the vibration isolation performance of the novel mechanical metamaterial under different payloads. In another study on chiral metamaterials for morphing wings’ applications, Genetic Algorithm (GA) proved a crucial component of the developed parametric design framework for gradient conformal chiral metamaterials. ¹⁹⁸ In this study, GA was utilised for automatic iterative optimisation, enabling the search for optimal combinations of gradient distribution parameters to enhance the wing's multi-directional stiffness and deformability for morphing wings. Through this, the GA helps to achieve specific improvements in tail stiffness performance and overall enhanced deformability, which are vital for practical morphing wing applications.

In an article by Ha et al., ¹⁸⁹ a generative machine learning methodology utilising generative inverse and surrogate forward neural network (NN) models was used to undertake rapid inverse design of metamaterials consisting of architectural cells with cubic symmetry. This approach enabled the creation of nearly all possible uniaxial compressive stress-strain curve cases while accounting for process-dependent errors from printing. The study reported that mechanical behaviour with full tailorability could be achieved with nearly 90% accuracy; that is, high prediction accuracy was achieved between the experimentally tested curves and target (desired) curves of each tailored section. In another article, Hu et al. ¹⁹⁹ used machine learning (ML) algorithms, specifically K-Nearest Neighbour (KNN) for classification and ANN for regression, combined with a non-dominated sorting genetic algorithm (NSGA-II), to optimise both the mechanical and acoustic performances in shell-based lattice metamaterials, particularly focusing on G-type TPMS (Gyroid) lattice structures.

As evidenced by the preceding publications, the power of ML in the design of metamaterials cannot be underscored as it optimises and predicts performance with high accuracy. It is evident that ML accelerates the metamaterial design process, optimises the design, and enhances the development of novel metamaterial structures for high-performance applications. We predict an increasing trend in the adoption of these techniques in the design of new metamaterial structures with the increasing demand for lightweight and high-strength components in aerospace and related industries.

Numerical simulations

Numerical studies on metamaterials are essential in predicting their properties and understanding their complex interactions with different stimuli. Finite element modelling (FEM) is the most common numerical tool for studying metamaterial behaviour. ²⁰⁰ In the published literature, numerical modelling through FEM has been used to predict stress distribution and deformation, ²⁰¹ calculation of key mechanical properties such as Young's Modulus, ²⁰² Poisson's Ratio, shear modulus, and fracture toughness.^116,203 It has also been used to optimise the geometry of the mechanical metamaterials to achieve the performance criterion by varying the structure parameters in the finite element tools. ²⁰⁴ Notably, finite element tools have become sophisticated and powerful to undertake defect analysis in metamaterials and enhance the accuracy of property predictions. For instance, Vera-Rodríguez et al. ²⁰⁵ introduced a methodology to simulate the impact of typical defects, prevalent in Selective Laser Melting (SLM), on the effective mechanical response of metamaterials using finite element modelling. They revealed that these defects, even under small elastic strains, induce substantial changes in the mechanical performance of the metamaterials, and the magnitude of these changes is dependent on both the specific metamaterial unit cell geometry and the nature and severity of the defects. Similarly, Ziemke et al. ²⁰⁶ presented a finite-element model to demonstrate the relationship between stochastic (Weibull) distributions of defects and effective lattice properties. A finite element modelling approach that enables the incorporation of a variety of shapes and surface defects directly into the finite element meshes of strut-based lattices was introduced by Echeta et al. ²⁰⁷ Defect prediction and analysis is an exciting area of research in this field due to the limited publications available as well as the complexity involved in the development of such models.

4D/reconfigurable Mechanical metamaterials

Metamaterials fabricated from stimulus-responsive materials are known as 4D metamaterials. Such structures can reconfigure automatically (in terms of shape or property) in response to an external stimulus. Some of the common stimuli-responsive materials include shape memory polymers and alloys, liquid crystal elastomers, and hydrogels. ²⁰⁸ The 4D metamaterials are manufactured via a combination of 3D printing with 4D printing, in which the stimuli-responsive materials are processed via 3D printing techniques; fused filament fabrication (FFF), direct ink writing (DIW), digital light processing (DLP), and powder bed fusion (PBF), such as selective laser melting (SLM) and electron beam melting (EBM). ²⁰⁹ The 4D mechanical metamaterials have potential for applications in diverse fields such as biomedical devices, energy absorption, and reconfigurable antennas. Despite the advances in the field of 4D printing, the subject is in its early stages, and the key challenges include: (1) the lagging application of multifunctional 3D printing technologies (e.g., coaxial DIW, gradient printing, ultrasound-assisted printing), (2) limitations in high-precision printing for micro/nano-scale structures with diverse materials, and (3) the need for enhanced post-processing research to mitigate issues like thermal and residual stresses affecting manufacturing accuracy. To track the state-of-the-art progress in this field, we summarise some of the key publications for the past two years in Table 8. Accordingly, digital light processing 3D printing is the most adopted technology in fabrication of 4D mechanical metamaterials due to its high accuracy. ²¹⁰

Table 8.

Some of the latest key publications on 4D printing of mechanical metamaterials.

Title/Reference	Response-stimuli materials	3D technology	Mechanical response
4D printing of customizable and reconfigurable mechanical metamaterials 211	-PLA (Polylactic Acid): Used as a high-stiffness, hard-phase material for structural support and load-bearing capacity. -TPU (Thermoplastic Polyurethane): Used as a high-flexibility, soft-phase material that contributes to damping performance.	Multi-material fused deposition modelling (FDM) 3D printer	-Customisable stiffness characteristics: The metamaterials exhibit versatile stiffness behaviours, i.e., positive, zero, and negative stiffness. -Enhanced Dynamic Performance: The metamaterials show significant capabilities in both cushioning (impact absorption) and vibration isolation. -Reprogrammable stiffness: Leveraging the shape memory properties enabled by 4D printing, the metamaterials can be reconfigured to achieve reprogrammable stiffness.
4D Printing of Chiral Mechanical Metamaterials with Modular Programmability using Shape Memory Polymer 212	−B80-P20 resin: A relatively soft material with a glass transition temperature (Tg) of 56°C, showing consistent shape memory performance when subjected to temperature changes (e.g., between 0°C and 100°C). -S-701 resin: A stiffer material with a glass transition temperature of 121°C. It is used for rigid components like the top and bottom disks and connecting rods to ensure structural integrity.	A self-built multi-material Digital Light Processing (DLP) printing system.	-Compression-Twist Coupling: The fundamental design of the chiral units allows for compression-twist coupling, meaning an out-of-plane compression induces an in-plane twisting behaviour. -The modular design, using discrete chiral building blocks, enables on-demand re-programmability of structural heterogeneity and chirality. -The shape memory effect of the B80-P20 resin allows for temporary shape locking after mechanical compression at a high temperature (e.g., 80°C) and subsequent cooling. The original shape can be recovered by reheating. This process can be performed in repeated cycles. -Tunable Poisson's Ratio: The assembled metamaterials demonstrate the ability to achieve tunable Poisson's ratio responses, including negative Poisson's ratio (NPR), positive Poisson's ratio (PPR), and zero Poisson's ratio (ZPR).
Multifunctional and reprogrammable 4D pixel mechanical metamaterials 213	Shape Memory Polymer (SMP), UV light curing resin. This SMP is activated by heat (temperature), with a glass transition temperature (Tg) of 50 °C, allowing for shape programming and recovery.	Digital Light Processing (DLP) 3D printing technique	-The 4D pixel metamaterials exhibit multifunctional and actively reprogrammable mechanical properties, which are achieved through heat-triggered transformations. -They demonstrate excellent shape memory (up to 99% recovery) and mechanical stability, alongside precisely tunable Poisson's ratio (positive to negative) and adjustable stiffness through internal geometry reprogramming. -This reconfigurable response enables applications in adaptive devices, as showcased by a reprogrammable gripper capable of adapting to various objects for soft robotics.
4D Thermomechanical metamaterials for soft microrobotics	Commercial photoresist SU-8 (a negative epoxy-based polymer)	Two-photon lithography (TPL)	-The microrobots demonstrate precise, controlled, and reversible motion in response to thermal stimuli, exhibiting excellent stability for over 1000 cycles at 1 Hz without degradation. -Composed of distinct rotational and translational elements, the microrobot can achieve a controlled 360° rotation and a 120 µm translational motion. -Furthermore, it can generate substantial force (e.g., 200 µN) to transport objects, successfully performing micro-assembly tasks such as mounting a micro-gear onto a shaft.
Tuneable mechanical performance and reusability of 4D-printed heterogeneous metamaterials using shape memory biomass-derived polymer 214	Biomass-derived poly(lactic acid) (PLA)-based Shape Memory Polymer (SMP)	Fused deposition modelling	-Tunable Stiffness and Energy Absorption: The elastic modulus was tuned from 30.6 MPa to 109.8 MPa, and the specific energy absorption (SEA) reached a maximum of 3.8 J/g, depending on the design. -High Reusability and Shape Recovery: The structure demonstrates a high shape recovery ratio (SRR) of 98.7% after being deformed and subsequently heated. -Controllable Deformation Mechanisms: Different heterogeneous designs lead to distinct deformation mechanisms under compression, which are crucial for achieving the desired tunable mechanical performance.
Development of strong, stiff and lightweight compression-resistant mechanical metamaterials by refilling tetrahedral wireframes 215	Poly(methyl methacrylate) (PMMA) resin	Digital Light Processing (DLP) 3D printing	-Refilling the DLP 3D-printed PMMA tetrahedral wireframes substantially increases their compressive strength and Young's modulus without significantly increasing their relative density. -PMMA refilling: Compressive strength increased by 64.9% and Young's modulus by 173.3% at a relative density of 10.7%. -Epoxy refilling: Compressive strength increased by 93.6% and Young's modulus by a remarkable 360% at the same relative density. -Improved deformation behaviour: The refilled metamaterials exhibit a more uniform and widespread deformation under compression compared to non-refilled wireframes.
Shape recovery properties and load-carrying capacity of a 4D printed thick-walled kirigami-inspired honeycomb structure 216	Commercially available UV-curable shape memory polymer (SMP) resin	A Desktop-based Digital Light Processing (DLP) 3D printer	-Excellent Shape Recovery and Programmability: The 4D printed structures demonstrate a high shape recovery ratio (SRR) of 98.6% for thick-walled designs. -Enhanced Load-Carrying Capacity: For instance, a 2 × 2 × 2 sample achieved an average load-carrying capacity of 38.65 N. -Controllable Deformation and Large Volume Change: They have a volume ratio of up to 4.7 between the recovered and original states.
SMP-based anisotropic chiral metamaterials with tunable mechanical properties and band gap characteristics 140	A photosensitive Shape Memory Polymer (SMP) resin	Digital Light Processing (DLP) 3D printer	-Tunable Anisotropic Mechanical Properties: Their Young's modulus and Poisson's ratio can be precisely tuned along different directions by varying geometric parameters. -Shape Memory and Reconfigurability: The metamaterials have a shape recovery rate of 98.6%. -Deformation: The unique chiral design leads to distinct deformation behaviours under axial compression and torsion.
A Bio-inspired 3D metamaterials with chirality and anti-chirality topology fabricated by 4D printing 217	Photosensitive resin	Photocuring 3D printer (Anycubic Photon)	-Tunable Poisson's Ratio: The metamaterials exhibit a tunable Poisson's ratio, capable of ranging from positive to negative values by strategically combining chiral and anti-chiral unit cells. -High Shape Memory and Reprogrammability of 98%. -Their Young's modulus, compressive strength, and overall energy absorption properties can be significantly modulated by adjusting geometric parameters.
Notch Flexure as Kirigami Cut for Tunable Mechanical Stretchability towards Metamaterial Application 218	Polyethene terephthalate (PET)	Laser cutting	-Tunable Mechanical Stretchability and 3D Pattern Formation -Adjustable Trigger Conditions -Negative Poisson's Ratio
Design of bioinspired bone-based interpenetrating metamaterial with tailored mechanical properties 219	Thermoplastic polymer, PA12 (Polyamide 12)	Selective Laser Sintering (SLS)	-Under quasi-static compression, the structures achieved a maximum specific energy absorption to maximum transmitted strength (SEA/MTS) of 3.9 J/g*MPa. Structures with linear and quadratic gradients showed a higher energy absorption capacity under greater loads. However, combining strut and sheet-based cellular structures was found to significantly reduce the SEA.

Outlook of research in mechanical metamaterials

As mentioned, mechanical metamaterials are increasingly finding applications in the biomedical field (for the fabrication of orthopaedic implants and flexible medical devices), ²²⁰ energy absorption and impact mitigation (for automotive safety-collision buffers, protective equipment, e.g., helmets, body armour, etc.), vibration and noise control (in aircraft, automobiles, and machinery), structural applications such as aerospace and automotive (due to high-to-stiffness ratios, impact absorption and crashworthiness applications) and civil engineering (for seismic resistance improvement and structural stability). ²²¹ Better-performing metamaterials are going to be designed and tested for these applications in future research.

In biomedical applications, metamaterials shall be designed to mimic the bone mechanical properties for customisable bone implants, leading to more durable and effective implants (see Figure 46). Promising results on porous auxetic CoCrMo bone scaffolds fabricated via laser-powder bed fusion were reported by Wanniarachchi and co-authors.^63,64 The concept of self-aware implants for a new generation of multifunctional metamaterial implantable devices is a promising direction for future research. Such structures will incorporate self-sensing, self-powering, and mechanical tunability features to enhance their performance as implants. The development of personalised and patient-specific implants to match the unique anatomy of the patient and enhance the durability of the implant. The development of fabric-auxetic composite structures for the fabrication of knee sleeves for sports applications will be a future focus of research (see Figure 47). ²²² The structures are expected to harness the negative Poisson's ratio characteristics of the auxetic metamaterials. They may also be used for the fabrication of bandages with excellent conformability and adhesion for wound healing (see Figure 47(c)). ²²³

Figure 46.

AI-generated image depicting future metamaterials mimicking bone structure for customised hard tissue implants.

Figure 47.

(a) AI-generated auxetic honeycomb metamaterial for knee sleeve applications, (b) showing sleeves for sports applications (obtained from pinterest.com), and (c) metamaterial-based bandages with high conformability (reused with permission from Elsevier Ltd).

There is a growing need for lightweight materials in the automotive industry to enhance safety and reduce fuel consumption. The focus emphasis shall be put on crash box systems in which NPR metamaterials will be explored for higher energy absorption for frontal impact mitigation (Figure 48). The research on the development of metamaterials for bumpers to improve sound, vibration, and harshness performance. There shall also be a focus on the development of safer seat belts using auxetics.

Figure 48.

Metamaterials developed for (a) car crash box ²²⁴ and (b) bumper (ai-generated images).

The utilisation of mechanical metamaterials is an emerging area in the design of smart structures and has the potential to shift the paradigm of soft robots towards intelligent machines. Ongoing research explores the design of a bio-inspired soft mechanical metamaterial using an auxetic structure. As an example, studies have shown that the contraction of buckling-driven elastomeric metamaterial modules facilitates the design of a metamaterial caterpillar. ²²⁵ Such approaches introduce novel soft-robotic locomotion mechanisms that can be easily extended to other bioinspired systems. Similar works have tried to overcome the challenges of achieving steering motion in soft robots by mimicking caterpillars’ bending control. The use of origami units as the building blocks for soft robots (mimicking segmented caterpillar bodies) has been demonstrated. ²²⁶ It has been suggested that the concept of modular soft robots can provide insight into future designs that may have the ability to grow, repair, and enhance functionality. Future research could expand on this by developing more sophisticated modular designs and actuation methods and exploring applications in diverse and complex environments.

The application of Artificial Intelligence (AI) and Machine Learning (ML) techniques, including neural networks and evolutionary algorithms, is increasingly prevalent in metamaterials engineering, as these computational methods offer reliable designs and predictions of the behaviour. The capacity of AI algorithms to efficiently manage the complexity and numerous variables involved in metamaterial design is a key advantage, enabling faster design processes than traditional trial-and-error or analytical approaches. ¹⁷⁸ Besides speeding up the development of metamaterials, ML facilitates the discovery of previously unknown correlations between design parameters and performance characteristics. Ultimately, AI and ML are crucial in driving the innovation of metamaterials, automating design and discovery while revealing latent patterns that may otherwise go unrecognised. According to the published literature, the use of AI and ML in designing metamaterials is expected to rise, as these technologies can automate and accelerate the design process. It is anticipated that AI techniques, including deep learning, will enhance the discovery of new metamaterials and their characteristics by spotting previously unnoticed patterns in data and uncovering latent physical laws. For example, machine learning algorithms can analyse large datasets of metamaterials’ properties and the geometrical configurations or structure of their unit cells to predict the properties of new metamaterials based on these structural features. ²²⁷ In mechanics, genetic algorithms have been utilised to optimise geometrical alterations in metamaterials to maximise their mechanical performance, and machine learning can predict mechanical parameters like strength and modulus. This involves creating models that are superior to traditional numerical analysis, which could lead to significant savings in computational resources. Furthermore, the integration of AI is identified as a key development, facilitating the design of metamaterials, and in some cases, metamaterials are being developed to perform AI functions, demonstrating the bidirectional nature of AI and metamaterials.

Conclusions

A review of the state-of-the-art of mechanical metamaterials has been presented in the article. Mechanical metamaterials can be classified according to the structure/geometry of the unit cell, unique mechanical properties, functionality, response to mechanical stimuli, and scale level sizes of the unit cells. There is a lot of work on developing NPR and high-energy absorption metamaterials for mechanical applications. The emphasis of research on these topics is on the development of new structures and the enhancement of the performance of the known design. As such, a huge pool of metamaterials exhibiting these two properties has been developed. There are concerted efforts to establish failure mechanisms of metamaterials, and as highlighted in the article, existing theories of fracture/failure are being modified and evaluated for various metamaterials. Numerical simulation and machine learning are practical tools for designing high-performance metamaterials.

The future of mechanical metamaterials is promising, especially in exploring standards for fracture characterisation, topology optimisation techniques, machine learning in design, and new applications. It is expected that research on machine learning in the design of mechanical metamaterials will intensify due to the availability of computing power. Research should focus on adopting mechanical metamaterials in practical applications, especially in biomedical implants poised for significant impact. Establishing industrial evaluation standards analogous to bulk materials should be a key consideration in this case. Adopting metamaterials for smart applications such as soft robots and sensor fabrication should be emphasised in present and future research efforts.