In this study, by mimicking bone tissue adaptation, we introduce a hierarchical design approach to optimize the mechanical performance of auxetic metamaterials. Using a multi-material voxel-based 3D printer, we rationally position soft and hard phases at the voxel level based on a bone-inspired remodeling algorithm that maintains the homeostasis of strain energy density. This process introduces an additional length scale within the structure, leading to (1) a significant expansion of the envelope of achievable elastic properties, (2) more homogeneous strain energy distributions, and (3) up to 78 % stronger metamaterials compared to initial designs. Our results demonstrate that this bone-mimicking design approach enables the emergence of an intermediate length scale between the unit cell and voxel scales, which is responsible for the observed improvements in mechanical performance. Taken together, these findings highlight the potential of biomimetic remodeling for the rational design of stress-worthy, multiscale mechanical metamaterials that combine unusual elastic properties with high mechanical performance.

The development of high-fidelity three-dimensional (3D) tissue models can minimize the need for animal models in clinical medicine and drug development. However, physical limitations regarding the distances within which diffusion processes are effective impose limitations on the size of such constructs. That is because larger-size constructs experience necrosis, especially in their centers, due to the cells residing deep inside such constructs not receiving enough oxygen and nutrients. This hampers the sustained in vitro growth of the tissues which is required for achieving functional microtissues. To address this challenge, we used three types of 3D printing technologies to create perfusable networks at different length scales and integrate them into such constructs. Toward this aim, networks incorporating porous conduits with increasingly complex configurations were designed and fabricated using fused deposition modeling, stereolithography, and two-photon polymerization while optimizing the printing conditions for each of these technologies. Furthermore, following network embedding in hydrogels, contrast agent-enhanced micro-computed tomography and confocal fluorescence microscopy were employed to characterize one of the essential network functionalities, namely the diffusion function. The investigations revealed the effects of various design parameters on the diffusion behavior of the porous conduits over 24 h. We found that the number of pores exerts the most significant influence on the diffusion behavior of the contrast agent, followed by variations in the pore size and hydrogel concentration. The analytical approach and the findings of this study establish a solid base for a new technological platform to fabricate perfusable multiscale 3D porous networks with complex designs while enabling the customization of diffusion characteristics to meet specific requirements for sustained in vitro tissue growth. Statement of significance: This study addresses an essential limitation of current 3D tissue engineering, namely, sustaining tissue viability in larger constructs through optimized nutrient and oxygen delivery. By utilizing advanced 3D printing techniques this research proposes the fabrication of perfusable, multiscale and customizable networks that enhance diffusion and enable cell access to essential nutrients throughout the construct. The findings highlighted the role of network characteristics on the diffusion of a model compound within a hydrogel matrix. This work represents a promising technological platform for creating advanced in vitro 3D tissue models that can reduce the use of animal models in research involving tissue regeneration, disease models and drug development.

High-performance soft–hard interfaces are inherently difficult to fabricate due to the dissimilar mechanical properties of both materials, especially when connecting extremely soft biomaterials, such as hydrogels, to much harder biomaterials, such as rigid polymers. Nevertheless, there is significant clinical demand for synthetic soft–hard interfaces. Here, soft–hard interface geometries are proposed, designed with the aid of computational analyses and fabricated as 3D-printed hydrogel-to-polylactide (PLA) structures. Two primary interlocking geometries (i.e., anti-trapezoidal (AT) and double-hook (DH)) are used to study the envelope of 2.5D geometric interlocking designs, fabricated through hybrid 3D printing, combining pneumatic extrusion with fused deposition modeling. Finite-element analysis, uniaxial tensile tests, and digital image correlation (DIC) are used to characterize the geometries and identify parameters that significantly influence their mechanical performance. These findings reveal significant differences between geometric designs, where DH geometries performed significantly better than AT geometries, exhibiting a 190% increase in the maximum force, F_max, and a 340% increase in the fracture toughness, W. Compared to the control groups (i.e., flat, inset, and 90° interfaces), F_max and W values increased by 500%–990% and 350%–1200%, respectively. The findings of this study can serve as a guideline for the design and fabrication of efficient soft–hard interfaces with performances close to predicted values.

This Roadmap surveys the diversity of different approaches for characterising, modelling and designing metamaterials. It contains articles covering the wide range of physical settings in which metamaterials have been realised, from acoustics and electromagnetics to water waves and mechanical systems. In doing so, we highlight synergies between the many different physical domains and identify commonality between the main challenges. The articles also survey a variety of different strategies and philosophies, from analytic methods such as classical homogenisation to numerical optimisation and data-driven approaches. We highlight how the challenging and many-degree-of-freedom nature of metamaterial design problems call for techniques to be used in partnership, such that physical modelling and intuition can be combined with the computational might of modern optimisation and machine learning to facilitate future breakthroughs in the field.

The authors regret that the affiliation of Joseph Ahn was incorrectly listed in the original article. The correct affiliation is:

Independent Researcher, Hwaseong-si, South Korea

This error has now been corrected in the online version of the article.

The authors would like to apologise for any inconvenience caused.

Emerging 4D printing techniques have enabled the realization of smart materials whose shape or properties can change with time. Two important phenomena play important roles in the 4D printing of shape memory polymeric materials. First, the anisotropic deformation of the printed filaments due to residual stresses can be harnessed to create out-of-plane shape transformations. Second, the unavoidable formation of micro-defects during the printing processes often affects the programmability of the printed object. Here, we propose a design approach that harnesses these two effects occurring during fused deposition modeling to create tailor-made curved geometries from initially 2D flat disks. We first determined the size and distribution of the imperfections formed within printed structures by varying two printing parameters namely the printing speed and the number of printed materials. Spatially varying the printing speed and combining polylactic acid filaments with a softer material without shape memory properties allowed us to cover a variety of shapes from negative to positive values of the mean and Gaussian curvature. We propose an analytical model to calculate the magnitude of the maximum out-of-plane deformation from the anisotropic expansion factor of the constituting microstructures. Furthermore, we develop computational models to predict the complex shape-changing of thermally actuated 4D printed structures given the distribution of rationally introduced imperfections and we demonstrate the potential applications of such defect-based metamaterials in drug delivery systems.

The Poisson's ratio and elastic modulus are two parameters determining the elastic behavior of biomaterials. While the effects of elastic modulus on the cell response is widely studied, very little is known regarding the effects of the Poisson's ratio. The micro-architecture of meta-biomaterials determines not only the Poisson's ratio but also several other parameters that also influence cell response, such as porosity, pore size, and effective elastic modulus. It is, therefore, very challenging to isolate the effects of the Poisson's ratio from those of other micro-architectural parameters. Here, we computationally design meta-biomaterials with controlled Poisson's ratios, ranging between -0.74 and +0.74, while maintaining consistent porosity, pore size, and effective elastic modulus. The 3D meta-biomaterials were additively manufactured at the micro-scale using two-photon polymerization (2PP), and were mechanically evaluated at the meso‑scale. The response of murine preosteoblasts to these meta-biomaterials was then studied using in vitro cell culture models. Meta-biomaterials with positive Poisson's ratios resulted in higher metabolic activity than those with negative values. The cells could attach and infiltrate all meta-biomaterials from the bottom to the top, fully covering the scaffolds after 17 days of culture. Interestingly, the meta-biomaterials exhibited different cell-induced deformations (e.g., shrinkage or local bending) as observed via scanning electron microscopy. The outcomes of osteogenic differentiation (i.e., Runx2 immunofluorescent staining) and matrix mineralization (i.e., Alizarin red staining) assays indicated the significant potential impact of these meta-biomaterials in the field of bone tissue engineering, paving the way for the development of advanced bone meta-implants. Statement of significance: We studied the influence of Poisson's ratio on bone cell response in meta-biomaterials. While elastic modulus effects are well-studied, the impact of Poisson's ratio, especially negative values found in architected biomaterials, remains largely unexplored. The complexity arises from intertwined micro-architectural parameters, such as porosity and elastic modulus, making it challenging to isolate the Poisson's ratio. To overcome this limitation, this study employed rational computational design to create meta-biomaterials with controlled Poisson's ratios, alongside consistent effective elastic modulus, porosity, and pore size. The study reveals that two-photon polymerized 3D meta-biomaterials with positive Poisson's ratios displayed higher metabolic activity, while all the developed meta-biomaterials supported osteogenic differentiation of preosteoblasts as well as matrix mineralization. The outcomes pave the way for the development of advanced 3D bone tissue models and meta-implants.

Take a thin cylindrical shell and twist it; it will buckle immediately. Such unavoidable torsional buckling can lead to systemic failure, for example by disrupting the blood flow through arteries. In this study, we prevent this torsional buckling instability using a combination of auxeticity and orthotropy in cylindrical metamaterial shells with a holey pattern. When the principal axes of the orthotropic meta-shell are relatively aligned with that of the compressive component of the applied stress during twisting, the meta-shell uniformly shrinks in the radial direction as a result of a local buckling instability. This shrinkage coincides with a softening-stiffening transition that leads to ordered stacking of unit cells along the compressive component of the applied stress. These transitions due to local instabilities circumvent the usual torsional instability even under a large twist angle. This study highlights the potential of tailoring anisotropy and programming instabilities in metamaterials, with potential applications in designing mechanical elements for soft robotics, biomechanics or fluidics. As an example of such applications, we demonstrate soft torsional compressor for generating pulsatile flows through a torsion release mechanism.

Temporomandibular joint (TMJ) replacement prostheses often face limitations in accommodating translational movements, leading to unnatural kinematics and loading conditions, which affect functionality and longevity. Here, we investigate the potential of functionally graded materials (FGMs) in TMJ prostheses to enhance mandibular kinematics and reduce joint reaction forces. We develop a functionally graded artificial cartilage for the TMJ implant and evaluate five FGM designs: hard, hard-soft, and three FGM gradients with gradual transitions from 90% hard material to 0%, 10%, and 20%. These designs are 3D printed, mechanically tested under quasi-static compression, and simulated under physiological conditions. Results from computational modeling and experiments are compared to an intact mandible during incisal clenching and left group biting. The FGM design with a transition from 90% to 0% hard material improves kinematics by 19% and decreases perfomance by 3%, reduces joint reaction forces by 8% and 10%, and increases mandibular movement by 20% and 88% during incisal clenching and left group biting, respectively. These findings provide valuable insights for next-generation TMJ implants.

The existing 3D printing methods exhibit certain fabrication-dependent limitations for printing curved constructs that are relevant for many tissues. Four-dimensional (4D) printing is an emerging technology that is expected to revolutionize the field of tissue engineering and regenerative medicine (TERM). 4D printing is based on 3D printing, featuring the introduction of time as the fourth dimension, in which there is a transition from a 3D printed scaffold to a new, distinct, and stable state, upon the application of one or more stimuli. Here, we present an overview of the current developments of the 4D printing technology for TERM, with a focus on approaches to achieve temporal changes of the shape of the printed constructs that would enable biofabrication of highly complex structures. To this aim, the printing methods, types of stimuli, shape-shifting mechanisms, and cell-incorporation strategies are critically reviewed. Furthermore, the challenges of this very recent biofabrication technology as well as the future research directions are discussed. Our findings show that the most common printing methods so far are stereolithography (SLA) and extrusion bioprinting, followed by fused deposition modelling, while the shape-shifting mechanisms used for TERM applications are shape-memory and differential swelling for 4D printing and 4D bioprinting, respectively. For shape-memory mechanism, there is a high prevalence of synthetic materials, such as polylactic acid (PLA), poly(glycerol dodecanoate) acrylate (PGDA), or polyurethanes. On the other hand, different acrylate combinations of alginate, hyaluronan, or gelatin have been used for differential swelling-based 4D transformations. TERM applications include bone, vascular, and cardiac tissues as the main target of the 4D (bio)printing technology. The field has great potential for further development by considering the combination of multiple stimuli, the use of a wider range of 4D techniques, and the implementation of computational-assisted strategies.

Meta-biomaterials, engineered materials with distinctive combinations of mechanical, physical, and biological properties stemming from their micro-architecture, have emerged as a promising domain within biomedical engineering. Correspondingly, meta-implants, which serve as the device counterparts of meta-biomaterials, offer exceptional functionalities, holding great potential for addressing complex skeletal diseases. This paper presents a comprehensive overview of the various types of meta-implants, including hybrid, shape-morphing, metallic clay, and deployable meta-implants, highlighting their unprecedented properties and recent achievement in the field. This paper also delves into the potential future developments of meta-implants, addressing the exploration of multi-functionalities in meta-biomaterials and their applications in diverse biomedical fields.

Thin-wall structures, particularly thin-walled tubes, play a critical role in load-bearing structures. Enhancing their ability to withstand crushing loads can significantly improve the overall damping efficiency of the system. Functionally graded thickness (FGT) is a promising approach for enhancing the load-bearing properties of thin-walled tubes by enabling control over material usage and localized deformation patterns within the structure. In this study, we proposed a novel theoretical model that analyzes the crushing behavior of hollow and foam-filled FGT thin-walled circular tubes by considering four primary failure mechanisms that contribute to energy dissipation: (1) bending of plastic hinges, (2) membrane stretching, (3) axial foam crushing, and (4) the interaction between foam and the tube's wall. We validated our model against experimental results from previous researchers and observed a good agreement. Additionally, we conduct a comprehensive study to examine the effects of various geometrical parameters, such as power-law functions and normalized wall thickness ratio, on the crushing behavior of FGT structures. Our results demonstrate the accuracy and reliability of our theoretical model and highlight the potential of FGT structures to enhance the performance of thin-walled tubes in a range of load-bearing applications.

The currently available treatments for inner ear disorders often involve systemic drug administration, leading to suboptimal drug concentrations and side effects. Cochlear implants offer a potential solution by providing localized and sustained drug delivery to the cochlea. While the mechanical characterization of both the implants and their constituent material is crucial to ensure functional performance and structural integrity during implantation, this aspect has been mostly overlooked. This study proposes a novel methodology for the mechanical characterization of our recently developed cochlear implant design, namely, rectangular and cylindrical, fabricated using two-photon polymerization (2 PP) with a novel photosensitive resin (IP-Q™). We used in silico computational models and ex silico experiments to study the mechanics of our newly designed implants when subjected to torsion mimicking the foreseeable implantation procedure. Torsion testing on the actual-sized implants was not feasible due to their small size (0.6 × 0.6 × 2.4 mm³). Therefore, scaled-up rectangular cochlear implants (5 × 5 × 20 mm³, 10 × 10 × 40 mm³, and 20 × 20 × 80 mm³) were fabricated using stereolithography and subjected to torsion testing. Finite element analysis (FEA) accurately represented the linear behavior observed in the torsion experiments. We then used the validated Finite element analysis models to study the mechanical behavior of real-sized implants fabricated from the IP-Q resin. Mechanical characterization of both implant designs, with different inner porous structures (pore size: 20 μm and 60 μm) and a hollow version, revealed that the cylindrical implants exhibited approximately three times higher stiffness and mechanical strength as compared to the rectangular ones. The influence of the pore sizes on the mechanical behavior of these implant designs was found to be small. Based on these findings, the cylindrical design, regardless of the pore size, is recommended for further research and development efforts.

4D (bio-)printing endows 3D printed (bio-)materials with multiple functionalities and dynamic properties. 4D printed materials have been recently used in biomedical engineering for the design and fabrication of biomedical devices, such as stents, occluders, microneedles, smart 3D-cell engineered microenvironments, drug delivery systems, wound closures, and implantable medical devices. However, the success of 4D printing relies on the rational design of 4D printed objects, the selection of smart materials, and the availability of appropriate types of external (multi-)stimuli. Here, this work first highlights the different types of smart materials, external stimuli, and design strategies used in 4D (bio-)printing. Then, it presents a critical review of the biomedical applications of 4D printing and discusses the future directions of biomedical research in this exciting area, including in vivo tissue regeneration studies, the implementation of multiple materials with reversible shape memory behaviors, the creation of fast shape-transformation responses, the ability to operate at the microscale, untethered activation and control, and the application of (machine learning-based) modeling approaches to predict the structure–property and design–shape transformation relationships of 4D (bio)printed constructs.

Shape morphing is the ability of objects to adapt to different shapes and reduce stress concentrations through increased contact area. This is a common trait of natural and engineered objects and has several applications in, among others, soft robotics and orthopedic implants. Shape morphing is achieved through flexible materials or rigid components with either kinematic or compliant joints. An additional step, namely shape locking, is needed for sustained load support. Activation of a locking mechanism can be done with any energy, among which magnetism is one. Here, we present the implementation of a magnetic locking mechanism for kinematically deformable metamaterial structures that maintain shape and support loads upon locking. The structure consists of 3D printed rigid magnetic and non-magnetic components connected by hinges. We created several prototypes of the proposed designs using two additive manufacturing methods (i.e., material extrusion and multi-material jetting) and demonstrated its application in a closed-loop grid for arbitrary shapes. Moreover, we characterized the performance of the prototypes using mechanical tests and multibody kinematic system simulations. This work highlights the viability of the locking concept and provides design considerations for future applications. Further improvement and optimizations are needed for increased efficiency and effectiveness.

Bone-to-soft tissue interfaces are responsible for transferring loads between tissues with significantly dissimilar material properties. The examples of connective soft tissues are ligaments, tendons, and cartilages. Such natural tissue interfaces have unique microstructural properties and characteristics which avoid the abrupt transitions between two tissues and prevent formation of stress concentration at their connections. Here, we review some of the important characteristics of these natural interfaces. The native bone-to-soft tissue interfaces consist of several hierarchical levels which are formed in a highly specialized anisotropic fashion and are composed of different types of heterogeneously distributed cells. The characteristics of a natural interface can rely on two main design principles, namely by changing the local microarchitectural features (e.g., complex cell arrangements, and introducing interlocking mechanisms at the interfaces through various geometrical designs) and changing the local chemical compositions (e.g., a smooth and gradual transition in the level of mineralization). Implementing such design principles appears to be a promising approach that can be used in the design, reconstruction, and regeneration of engineered biomimetic tissue interfaces. Furthermore, prominent fabrication techniques such as additive manufacturing (AM) including 3D printing and electrospinning can be used to ease these implementation processes. Biomimetic interfaces have several biological applications, for example, to create synthetic scaffolds for osteochondral tissue repair.

Objective: Mandibular reconstruction using patient-specific cage implants is a promising alternative to the vascularized free flap reconstruction for nonirradiated patients with adequate soft tissues, or for patients whose clinical condition is not conducive to microsurgical reconstruction. This study aimed to assess the biomechanical performance of 3D printed patient-specific cage implants designed with a semi-automated workflow in a combined cadaveric and retrospective case series study. Methods: We designed cage implants for two human cadaveric mandibles using our previously developed design workflow. The biomechanical performance of the implants was assessed with the finite element analysis (FEA) and quasi-static biomechanical testing. Digital image correlation (DIC) was used to measure the full-field strains and validate the FE models by comparing the distribution of maximum principal strains within the bone. The retrospective study of a case series involved three patients, each of whom was treated with a cage implant of similar design. The biomechanical performance of these implants was evaluated using the experimentally validated FEA under the scenarios of both mandibular union and nonunion. Results: No implant or screw failure was observed prior to contralateral bone fracture during the quasi-static testing of both cadaveric mandibles. The FEA and DIC strain contour plots indicated a strong linear correlation (r = 0.92) and a low standard error (SE=29.32με), with computational models yielding higher strain values by a factor of 2.7. The overall stresses acting on the case series’ implants stayed well below the yield strength of additively manufactured (AM) commercially pure titanium, when simulated under highly strenuous chewing conditions. Simulating a full union between the graft and remnant mandible yielded a substantial reduction (72.7±1.5%) in local peak stresses within the implants as compared to a non-bonded graft. Conclusions: This study shows the suitability of the developed semi-automated workflow in designing patient-specific cage implants with satisfactory mechanical functioning under demanding chewing conditions. The proposed workflow can aid clinical engineers in creating reconstruction systems and streamlining pre-surgical planning. Nevertheless, more research is still needed to evaluate the osteogenic potential of bone graft insertions.

This paper presents an analytical model to investigate the static behaviour of sandwich plates comprised of two isotropic face sheets and a honeycomb core. Through-thickness transverse shear stresses were considered using a unified displacement field with which various plate theories were implemented, i.e., exponential, third-order, hyperbolic, sinusoidal, fifth-order, Mindlin, and the classic plate theory. The equilibrium equations of a simply-supported sandwich panel were derived using the principle of virtual work and Navier solution was obtained under static transverse loading. After validating of the model, various mechanical and geometrical parameters were varied to characterise the behaviour of the structure under regular and auxetic response. It was found that the auxeticity of the core strongly affects the mechanical response, e.g., in controlling deflection, in-plane anisotropy, and Poisson’s ratio. Cell wall angle was found to be most critical parameter that can be used to adjust anisotropy, out-of-plane shear modulus, transverse shear stress distribution, and deflection of the panel. Also the cell aspect ratio controls the sensitivity of the core response to other geometrical variations. In terms of the higher-order theories, the deflection-dependent parameter of the unified formulation seems to have more control of maximum deflection compared to independent rotations. Auxeticity of the core showed some benefits in controlling anisotropy, deflection and providing additional out-of-plane shear rigidity. Overall, since there is not one-to-one relationship between specific values of Poisson’s ratio, anisotropy, and shear rigidity, careful design considerations must be invested to obtain a correct mechanical response.

Practical applications of mechanical metamaterials often involve solving inverse problems aimed at finding microarchitectures that give rise to certain properties. The limited resolution of additive manufacturing techniques often requires solving such inverse problems for specific specimen sizes. Moreover, the candidate microarchitectures should be resistant to fatigue and fracture. Such a multi-objective inverse design problem is formidably difficult to solve but its solution is the key to real-world applications of mechanical metamaterials. Here, a modular approach titled “Deep-DRAM” that combines four decoupled models is proposed, including two deep learning (DL) models, a deep generative model based on conditional variational autoencoders, and direct finite element (FE) simulations. Deep-DRAM integrates these models into a framework capable of finding many solutions to the posed multi-objective inverse design problem based on random-network unit cells. Using an extensive set of simulations as well as experiments performed on 3D printed specimens, it is demonstrate that: 1) the predictions of the DL models are in agreement with FE simulations and experimental observations, 2) an enlarged envelope of achievable elastic properties (e.g., rare combinations of double auxeticity and high stiffness) is realized using the proposed approach, and 3) Deep-DRAM can provide many solutions to the considered multi-objective inverse design problem.

This broad review summarizes recent advances and "hot"research topics in nanophononics and elastic, acoustic, and mechanical metamaterials based on results presented by the authors at the EUROMECH 610 Colloquium held on April 25-27, 2022 in Benicássim, Spain. The key goal of the colloquium was to highlight important developments in these areas, particularly new results that emerged during the last two years. This work thus presents a "snapshot"of the state-of-the-art of different nanophononics- and metamaterial-related topics rather than a historical view on these subjects, in contrast to a conventional review article. The introduction of basic definitions for each topic is followed by an outline of design strategies for the media under consideration, recently developed analysis and implementation techniques, and discussions of current challenges and promising applications. This review, while not comprehensive, will be helpful especially for early-career researchers, among others, as it offers a broad view of the current state-of-the-art and highlights some unique and flourishing research in the mentioned fields, providing insight into multiple exciting research directions.