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Abstract
Recent advances in additive manufacturing make it possible to fabricate periodic lattice structures with complex configurations. However, a proper design strategy to achieve lattice structures with controlled anisotropy is still unavailable. There is an urgent need to fill this knowledge gap in order to develop mechanical metamaterials with prescribed properties. Here we propose two different methodologies to design lattice structures with controlled anisotropy. As examples, we created two new families of lattice structures with isotropic elasticity and cubic symmetric geometry. The findings of this work provide simple and effective strategies for exploring lightweight metamaterials with desired mechanical properties.
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... A compression load case to calculate the stiffness components C 11 , C 12 which relate to normal stresses and a shear load case for the calculation of C 44 which specifies the shear modulus for a structure as shown in Fig. 10a. This methodology has been previously employed for the precise calculation of the polar stiffness map of structure, leading to a close correlation between theoretical and experimental results 52 . Two types of boundary conditions for compression, Eq. (9), and shear testing, Eq. (10), are defined. ...
... Two types of boundary conditions for compression, Eq. (9), and shear testing, Eq. (10), are defined. Take a normal strain ε xx for instance, the boundary conditions were defined by 52 : ...
... Further information on the applied boundary conditions and FEA models can be found in the Supplementary Figure 1 and Figure 2 and elsewhere 46,52,65 . Utilizing the simulations and effective stiffness properties, the cost function values for isotropy and auxeticity can be determined. ...
Recent advancements in manufacturing, finite element analysis (FEA), and optimization techniques have expanded the design possibilities for metamaterials, including isotropic and auxetic structures, known for applications like energy absorption due to their unique deformation mechanism and consistent behavior under varying loads. However, achieving simultaneous control of multiple properties, such as optimal isotropic and auxetic characteristics, remains challenging. This paper introduces a systematic design approach that combines modeling, FEA, genetic algorithm, and optimization to create tailored mechanical behavior in metamaterials. Through strategically arranging 8 distinct neither isotropic nor auxetic unit cell states, the stiffness tensor in a 5 × 5 × 5 cubic symmetric lattice structure is controlled. Employing the NSGA-II genetic algorithm and automated modeling, we yield metamaterial lattice structures possessing both desired isotropic and auxetic properties. Multiphoton lithography fabrication and experimental characterization of the optimized metamaterial highlights a practical real-world use and confirms the close correlation between theoretical and experimental data.
... Isotropic materials exhibit the same mechanical properties in all directions, making them highly desirable for achieving uniform load distribution and structural stability. To quantify the variance for the stiffness in different directions of cubic structures, the Zener ratio A is used and defined as [51,52]: ...
... A compression load case to calculate the stiffness components C 11 , C 12 which relate to normal stresses and a shear load case for the calculation of C 44 which specifies the shear modulus for a structure as shown in Figure 10a. This methodology has been previously employed for the precise calculation of the polar stiffness map of structure, leading to a close correlation between theoretical and experimental results [52]. The elastic stiffness components can be calculated by dividing the average strain (ε) through the average stress (σ) [61][62][63]: ...
... Further information on the applied boundary conditions and FEA models can be found in the SI and elsewhere [46,52,63]. Utilizing the simulations and effective stiffness properties, the cost function values for isotropy and auxeticity can be determined. ...
Recent advancements in manufacturing, finite element analysis (FEA), and optimization techniques have expanded the design possibilities for metamaterials, including isotropic and auxetic structures, known for applications like energy absorption due to their unique deformation mechanism and consistent behavior under varying loads. However, achieving simultaneous control of multiple properties, such as optimal isotropic and auxetic characteristics, remains challenging. This paper introduces a systematic design approach that combines modeling, FEA, neural networks, and optimization to create targeted mechanical behavior in metamaterials. Through strategically arranging 8 distinct neither isotropic nor auxetic unit cell states, the stiffness tensor in a 5 × 5 × 5 cubic symmetric lattice structure is controlled. Employing the NSGA-II genetic algorithm and automated modeling, we yield metamaterial lattice structures possessing both, desired isotropic and auxetic properties. Multiphoton lithography fabrication and experimental characterization of the optimized metamaterial highlights a practical real-world use and confirms close correlation between theoretical and experimental data.
... Lumpe and Stankovic [8] investigated numerous pentamode structures showing the importance of crystallographic symmetries on the mechanical properties of the structure. Xu et al [9] proposed two different methodologies to design lattice structures with controlled anisotropy. Huang et al. [10] introduced a new class of pentamode metamaterial with centrosymmetry, showing better mechanical characteristics. ...
... Thus, for the vertical stiffness Eqs. (8) and (9) can be suggested for the equivalent diameter for SPMs and TPMs respectively: ...
... d eq = 0.09×D + 0.91×d (9) Similarities can be found between the response to the horizontal and the vertical loading. Once again, the thicker specimens seem to be strongly influenced by the variation of the diameters of the beams. ...
Metameterials have unique properties, which are mostly attributed to their geometrical configuration. Pentamodes, a subcategory of metamaterials, exhibit an almost zero shear elastic modulus while maintaining high compression stiffness, offering a behavior similar to that of a liquid, suggesting the potential application of pentamodes in seismic isolation. In this paper a real-life bridge bearing, composed of repetitive layers of pentamode unit cells in the horizontal and vertical axes is studied. The lattices are modelled using beam finite elements with an equivalent uniform diameter to ensuring a stiffness equal to that of the bi-cone rod. The importance of the chosen equivalent diameter is shown, as the assumption of an average diameter of the bi-cone may lead to significant discrepancies between the calculated stiffnesses. For small bi-cone diameters difference, and slender formulations, the error could grow up to 15% for the horizontal stiffness and up to 200% for vertical. For thick formulations the average diameter overestimates the horizontal stiffness by 3 times and the vertical by 4. These discrepancies grow exponentially as the bi-cone diameters difference increases. An elastoplastic material is selected. The bearing supporting the superstructure is subjected to a constant vertical weight load and a horizontal shear base load, due to seismic excitation. Under vertical loading plastic hinges are created in all the rods of the cell and bearing. However, under shear loading plastic hinges are rather initially created in the lowest nodes of the cell and the bearing.
... Such lattice metamaterials, widely discussed in publications [6,8,10,19,20], represent an area of significant research interest. Various forms of lattice unit cells are extensively researched, and they can be constructed from struts [21][22][23][24], plates [19,[25][26][27], or curved surfaces [28][29][30][31][32]. Of central interest is the determination of the homogenized elastic material properties of lattice structures. ...
... These include, for example, porous materials [42] and biological tissues in medical research [43,44]. In ref. [24], anisotropic strut lattice structures are considered. In ref. [19,27], anisotropic plate lattice metamaterials are investigated, and it is shown that they can be transformed into isotropic metamaterials through the adjustment of certain geometric parameters. ...
Additive manufacturing enables the production of lattice structures, which have been proven to be a superior class of lightweight mechanical metamaterials whose specific stiffness can reach the theoretical limit of the upper Hashin–Shtrikman bound for isotropic cellular materials. To achieve isotropy, complex structures are required, which can be challenging in powder bed additive manufacturing, especially with regard to subsequent powder removal. The present study focuses on the Finite Element Method simulation of 2,5D anisotropic plate lattice metamaterials and the investigation of their lightweight potential. The intentional use of anisotropic structures allows the production of a cell architecture that is easily manufacturable via Laser Powder Bed Fusion (LPBF) while also enabling straightforward optimization for specific load cases. The work demonstrates that the considered anisotropic plate lattices exhibit high weight-specific stiffnesses, superior to those of honeycomb structures, and, simultaneously, a good de-powdering capability. A significant increase in stiffness and the associated surpassing of the upper Hashin–Shtrikman bound due to anisotropy is achievable by optimizing wall thicknesses depending on specific load cases. A stability analysis reveals that, in all lattice structures, plastic deformation is initiated before linear buckling occurs. An analysis of stress concentrations indicates that the introduction of radii at the plate intersections reduces stress peaks and simultaneously increases the weight-specific stiffnesses and thus the lightweight potential. Exemplary samples illustrate the feasibility of manufacturing the analyzed metamaterials within the LPBF process.
... In the literature [5,[26][27][28], there might be specific titles for such geometries/branches presented next, although (and to better identify the final geometry combination) in the context of this work these were the titles given. Each geometry is identified as the arbitrary name of the outer branches plus arbitrary name of the inner branches. ...
... The sensitivity analysis to be performed shares an identical path to the one presented previously in Section 2.4, although now a third design variable has to be taken into consideration: the aspect ratio, ρ ext . As stated, Equations (28) and (29) express the additional derivatives of the volume constraints with regards to ρ ext , to properly conclude this step. ...
Lattice structures are becoming an increasingly attractive design approach for the most diverse engineering applications. This increase in popularity is mainly due to their high specific strength and stiffness, considerable heat dissipation, and relatively light weight, among many other advantages. Additive manufacturing techniques have made it possible to achieve greater flexibility and resolution, enabling more complex and better-performing lattice structures. Unrestricted material unit cell designs are often associated with high computational power and connectivity problems, and highly restricted lattice unit cell designs may not reach the optimal desired properties despite their lower computational cost. This work focuses on increasing the flexibility of a restricted unit cell design while achieving a lower computational cost. It is based on a two-scale concurrent optimization of the lattice structure, which involves simultaneously optimizing the topology at both the macro- and micro-scales to achieve an optimal topology. To ensure a continuous optimization approach, surrogate models are used to define material and geometrical properties. The elasticity tensors for a lattice unit cell are obtained using an energy-based homogenization method combined with voxelization. A multi-variable parameterization of the material unit cell is defined to allow for the synthesis of functionally graded lattice structures.
... Through additive manufacturing (AM), patient-specific or complex three-dimensional objects, such as lattice structures, can be created cost-effectively by adding materials layer by layer. Unlike traditional manufacturing methods, which involve removing material from a solid block or bringing the parts together to make up a whole, additive manufacturing creates an object from scratch and is well-suited for the production of lattice structures [8][9][10]. ...
... Previous studies have investigated various lattice structures that can be used for prosthetics. The focus was mainly on lattice shapes, load directions, strain distributions, the creation of porous devices by material extraction, and the study of stiffness matrices [9,12,[14][15][16]. Liu et al. investigated the effect of several different stem types and different auxetic designs on stress shielding and reported that although the results varied with stem type and Gruen zones, auxetic stems caused less stress shielding than their solid counterparts [17]. ...
Artificial hip joints are used to replace damaged or diseased natural joints. When the stress that is typically applied to the bone changes because the implant and bone are different in stiffness, a phenomenon known as stress shielding occurs. Stress shielding can lead to bone weakening through reduced density and aseptic loosening in the long term. Studies are ongoing to overcome this phenomenon through geometric design, the use of materials with a low modulus of elasticity, or latticed implants. In this study, the effect of lightening the hip prosthesis with lattice structures on stress shielding is investigated using finite element simulation. The femoral stem of a solid hip prosthesis was lightweighted, with a re-entrant honeycomb auxetic cellular lattice structure, and structural analysis was performed. Two different lattice orientations were used, and it was observed that the stress distribution was more homogeneous in both orientations. In these femoral stems, which can be easily produced using additive manufacturing methods, a volume reduction of up to 16% was achieved. The stress transmitted to the bone increased by more than 36%, depending on the orientation, which is a promising result for reducing the stress shield effect.
... Parametric design allows the user to define a group of geometric parameters, including unit size, geometry, permutation, and so forth. These parameters can then be varied to produce different lattice structures [15][16][17]. This method is flexible and easy to modify, but the range of selectable parameters is considerable. ...
The yield strength and Young’s modulus of lattice structures are essential mechanical parameters that influence the utilization of materials in the aerospace and medical fields. Currently, accurately determining the Young’s modulus and yield strength of lattice structures often requires conduction of a large number of experiments for prediction and validation purposes. To save time and effort to accurately predict the material yield strength and Young’s modulus, based on the existing experimental data, finite element analysis is employed to expand the dataset. An artificial neural network algorithm is then used to establish a relationship model between the topology of the lattice structure and Young’s modulus (the yield strength), which is analyzed and verified. The Gibson–Ashby model analysis indicates that different lattice structures can be classified into two main deformation forms. To obtain an artificial neural network model that can accurately predict different lattice structures and be deployed in the prediction of BCC-FCC lattice structures, the artificial network model is further optimized and validated. Concurrently, the topology of disparate lattice structures gives rise to a certain discrete form of their dominant deformation, which consequently affects the neural network prediction. In conclusion, the prediction of Young’s modulus and yield strength of lattice structures using artificial neural networks is a feasible approach that can contribute to the development of lattice structures in the aerospace and medical fields.
... Architecting the structure of metamaterials across scales allows for achieving properties and functionalities not found in classical engineering materials, significantly expanding the traditional design space (1)(2)(3)(4)(5)(6). Examples include periodic truss metamaterials with high stiffness-and strength-to-weight ratios (7,8), lattice microstructures with tunable negative Poisson's ratio (9,10), and programmable anisotropic stiffness (11). Traditionally, the design process has been guided by engineering intuition (12), nature-inspiration (13) or topology optimization (14), hinging on prior expertise or computationally intensive calculations. ...
Material responses to static and dynamic stimuli, represented as nonlinear curves, are design targets for engineering functionalities like structural support, impact protection, and acoustic and photonic bandgaps. Three-dimensional metamaterials offer significant tunability due to their internal structure, yet existing methods struggle to capture their complex behavior-to-structure relationships. We present GraphMetaMat, a graph-based framework capable of designing three-dimensional metamaterials with programmable responses and arbitrary manufacturing constraints. Integrating graph networks, physics biases, reinforcement learning, and tree search, GraphMetaMat can target stress-strain curves spanning four orders of magnitude and complex behaviors, as well as viscoelastic transmission responses with varying attenuation gaps. GraphMetaMat can create cushioning materials for protective equipment and vibration-damping panels for electric vehicles, outperforming commercial materials, and enabling the automatic design of materials with on-demand functionalities.
... The mismatch of anisotropy between bone and implant may lead to stress shielding effect. Unlike solid metal materials, which are generally isotropic at the macro level, the anisotropy of lattice structures usually changes with the arrangement of the struts [32]. The anisotropic requirement for bone implants depends on the anatomical location of the application, such as trabecular bone requiring isotropy, while cortical bone requires strong anisotropy along the local loading direction [4]. ...
Lattice structures to fabricate bone implants can avoid stress-shielding effects and promote bone-in-growth. However, the performance of bones varies in different body parts, creating a barrier to manufacture an appropriate lattice structure for bone implant. Here, the formability, anisotropy, energy absorption abilities, stress distribution, and deformation mode of the laser powder bed fusion (LPBF) processed face-centered cubic (FCC), Octet, and Kelvin lattice structures were systematically compared through experiments and finite element analysis. The results show that the Kelvin lattice structure had the optimal comprehensive mechanical performance. This research has potential value for the design and manufacturing of specific bone implants.
... Representation surfaces of Young's modulus are used as a method to evaluate the anisotropy of monocrystallines and lattice structures [11,17]. The surface is a spherical surface, with the length of each radius vector being the reciprocal of Young's modulus corresponding to the direction. ...
We employ a mathematical model to analyze stress chains in thermoplastic elastomers (TPEs) with a microphase-separated spherical structure composed of triblock copolymers. The model represents stress chains during uniaxial and biaxial extensions using networks of spherical domains connected by bridges. We advance previous research and discuss permanent strain and other aspects of the network. It explores the dependency of permanent strain on the extension direction, using the average of tension tensors to represent isotropic material behavior. The concept of deviation angle is introduced to measure network anisotropy and is shown to play an essential role in predicting permanent strain when a network is extended in a specific direction. The paper also discusses methods to create a new network structure using various polymers.
... The simple cubic (SC) architecture of "boxes" has been utilized as all struts, all plates, and hollow struts filled with materials of varying moduli with arrangement to produce isotropic performance [14]. The body-centered cubic (BCC) and the face-centered cubic (FCC) have also been extensively investigated [15][16][17][18]. The FCC lattice, particularly due to the 12 struts emanating from a single node, has been further investigated as octet and tetrahedral lattices. ...
The utilization of lattice-type cellular architectures has seen a significant increase, owing to their predictable shape and the ability to fabricate templated porous materials through low-cost 3D-printing methods. Frames based on atomic lattice structures such as face-centered cubic (FCC), body-centered cubic (BCC), or simple cubic (SC) have been utilized. In FDM, the mechanical performance has been impeded by stress concentration at the nodes and melt-solidification interfaces arising from layer-by-layer deposition. Adding plates to the frames has resulted in improvements with a concurrent increase in weight and hot-pocket-induced dimensional impact in the closed cells formed. In this paper, we explore compressive performance from the partial addition of plates to the frames of a SC-BCC lattice. Compression testing of both single unit cells and 4 × 4 × 4 lattices in all three axial directions is conducted to examine stress transfer to the nearest neighbor and assess scale-up stress transfer. Our findings reveal that hybrid lattice structure unit cells exhibit significantly improved modulus in the range of 125% to 393%, specific modulus in the range of 13% to 120%, and energy absorption in the range of 17% to 395% over the open lattice. The scaled-up lattice modulus increased by 8% to 400%, specific modulus by 2% to 107%, and energy absorption by 37% to 553% over the lattice frame. Parameters that emerged as key to improved lightweighting.
... To design metastructures with controllable anisotropy, Xu et al. (Xu et al., 2016) have proposed two methods: (1) assembling different inner and outer lattice unit cells, and (2) tuning the rod diameter ratio of the inner and outer lattice unit cells. These methods can produce mechanical metastructures with varying anisotropic properties. ...
Anisotropic mechanical metamaterials with controllable properties are crucial for additive manufacturing design. However, manually regulating microstructural anisotropy remains challenging. This study introduces a method for artificial intelligence-aided design (AIAD) and mechanical metastructures optimization (MMO) to achieve extensive multi-scale structural enhancements. The approach involves compiling a comprehensive database of lattice materials with anisotropic characteristics. This is achieved by manipulating the central node position and rod diameter of a cubic-BCC microstructure. Homogenization theory then determines the elastic tensor of each microstructure. A 3D convolutional neural network (3D-CNN) maps the relationship between geometric properties and mechanical performance. An inverse design model based on a backpropagation neural network (NN) and parametric design acquires microstructures with desired elastic tensor attributes. Finally, a novel optimization approach for large-scale multiscale structures applies this method to control structural anisotropy. The resulting material distribution resembles a truss, significantly improving structural performance.
Keywords: artificial intelligence-aided; inverse design; neural network; metastructures optimization; controllable anisotropy
... The piecewise linear trend of Young's modulus previously found is confirmed also for the non-zero coefficients E11, E12 and E44. It is known that continuum metal materials are macroscopically isotropic due to the random distribution of crystalline grains, instead, lattice structures exhibit a commonly elastic anisotropy (Xu et al., 2016;Dong et al., 2019). In the present study the Zener coefficient (Zr) has been assessed to measure the anisotropy of the structure. ...
Rapid prototyping (RP) technology enables the fabrication of complex geometries, making lattice structures increasingly popular. Lattice structures, known as cellular materials, have garnered significant attention over the past two decades due to their ability to optimise mass distribution in components. These structures excel in mechanical properties, catering to energy absorption (bending-dominated structures) and structural performance (stretch-dominated structures). In this paper, we investigate the behaviour of stretch-dominated lattice structures using periodic surface models, specifically focusing on sheet-based Gyroid cells, to allow for a more efficient macroscale modelling. We study cells and scaffolds of different sizes, considering various triply periodic minimal surface thicknesses and relative densities ranging from approximately 0.2 to 0.65. We explore load applications in directions different from the unit cell's principal axes and analyse the strain rate effect on both bulk and cellular material. The lattice structures are manufactured using epoxy resin and digital light processing (DLP) technology. In the range of relative density investigated, both in quasi-static and dynamic conditions, a linear trend is observed for Young's modulus and compression yield strength. To extend the quasi-static results to the dynamic regime, we employ a more generalized normalization technique. This approach divides Young's modulus and compression yield strength by the behaviour of the base material at a specific strain rate, facilitating the correlation of mechanical properties across the two loading regimes. Based on experimental findings, we implemented and calibrated a bi-linear material model for describing, in macroscale, triply periodic minimal surface (TPMS) Gyroid structures. The model coefficients are parameterized with respect to relative density. In addition, the presented material law was compared with that proposed by Gibson-Ashby. Furthermore, we evaluated the anisotropy of both the base material and the unit cell. The first one is done by testing the 3D printed samples in directions different from the printing one, the latter by using the Zener factor. The anisotropy evaluation confirmed the isotropic behaviour of the unit cell within the range of relative density and test conditions investigated. Finally, we perform linear elastic 3D macroscopic and mesoscopic model simulations for combined shear-compression tests using the implemented bi-linear material model and the anisotropic stiffness matrix (obtained through the homogeneous formulation) for the macroscale, and the base material for the mesoscopic one. The results demonstrate the suitability of the proposed equivalent material model for studying the TPMS Gyroid structure in the elastic regime, both in quasi-static and dynamic states. This allows for an efficient FE modelling process of complex lattice structures.
... Auxetics also can be called mechanical metamaterials. Auxetic metamaterials have the following improved mechanical properties: (a) in-plane indentation resistance 10,11 ; (b) shear resistance [12][13][14] ; (c) synclastic behavior 15,16 ; (d) energy absorption [17][18][19][20][21][22][23][24][25][26][27][28] ; (e) fracture toughness [29][30][31] ; (f) negative compliance [32][33][34] ; and (g) sound insulation. [35][36][37] A novel auxetic honeycomb was produced with enhanced inplane crushing strength. ...
In this study, polylactic acid (PLA) and thermoplastic polyurethane (TPU) were used to fabricate composite materials with different raster orientations by a 3D printing process. The idea of this study came from the fact that in nature, many plants have vertical or horizontal lines on their leaves or branches. The authors made use of the composite material of a material structure under a thermal effect to achieve the material concept. Finally, their characteristics were observed by applying a thermal effect, tensile tests (to evaluate toughness), and a dynamic mechanical analysis (tan δ to evaluate energy absorption). After a thermal effect (temperature) from a heat source was applied, it was found that the material exhibited negative Poisson's ratio (NPR) characteristic. A single material with two layers on 0 ⁰ raster orientation of a material in this study had the maximum tan δ (energy absorption), an NPR, and toughness regardless of the kind of material. Among the composite materials, 0° raster orientation of them also had a larger NPR, and the absolute value of Poisson's ratio decreased as the temperature increased.
Highlights
The innovative two‐layer composite material fabricated by 3D printing.
The material characteristics with various raster orientations are discussed.
PLA0 has the largest tan δ , toughness, and negative Poisson's ratio.
... The elastic modulus provides valuable insights into mechanical compatibility with human bone [38]. Moreover, considering elastic anisotropy in scaffold design enables more efficient stress distribution, resulting in enhanced load-bearing capacity under complex loading conditions [39]. Additionally, the local stress distribution is a critical indicator of failure behaviors and should be duly considered [40]. ...
The present study investigated the influence of pore size of strut-based Diamond and surface-based Gyroid structures for their suitability as medical implants. Samples were made additively from laser powder bed fusion process with a relative density of 0.3 and pore sizes ranging from 300 to 1300 μm. They were subsequently examined for their manufacturability and mechanical properties. In addition, non-Newtonian computational fluid dynamics and discrete phase models were conducted to assess pressure drop and cell seeding efficiency. The results showed that both Diamond and Gyroid had higher as-built densities with smaller pore sizes. However, Gyroid demonstrated better manufacturability as its relative density was closer to the as-designed one. In addition, based on mechanical testing, the elastic modulus was largely unaffected by pore size, but post-yielding behaviors differed, especially in Diamond. High mechanical sensitivity in Diamond could be explained partly by Finite Element simulations, which revealed stress localization in Diamond and more uniform stress distribution in Gyroid. Furthermore, we defined the product of the normalized specific surface, normalized pressure drop, and cell seeding efficiency as the indicator of an optimal pore size, in which this factor identified an optimal pore size of approximately 500 μm for both Diamond and Gyroid. Besides, based on such criterion, Gyroid exhibited greater applicability as bone scaffolds. In summary, this study provides comprehensive assessment of the effect of pore size and demonstrates the efficient estimation of an in-silico framework for evaluating lattice structures as medical implants, which could be applied to other lattice architectures.
... The mechanical properties of cellular structures are influenced by various factors, including the material properties, manufacturing process, cellular unit geometry, and arrangement of units. 6,[10][11][12][13][14] With the development of 3D printing technology, various cellular structures with low density and high damping properties can now be fabricated and utilized in many fields. Cellular structures can be categorized into three main types based on the closure property of the cellular units: open-cell, hybrid-cell, and closed-cell structures. ...
In recent years, significant research has been conducted to explore the use of 3D triply periodic minimal surface (TPMS) structures for their exceptional vibrational damping properties and their ability to provide a continuous, smooth surface. The emergence of 3D printing has enabled the application of TPMS structures in fields such as medicine and aviation. In civil engineering, the compressive capacity of structures is a fundamental parameter in structural design. To evaluate the potential of porous TPMS structures in civil engineering, we have designed and manufactured four types of Skeletal-TPMS units using Stereolithography (SLA) technology. Axially loaded tests and nonlinear finite element method (NFEM) simulations have been performed to investigate the compressive strength and stiffness of the units. Our findings indicate that compared to solid blocks, the compressive strength of Skeletal-TPMS units decreases by 71.3% to 82.6%, and the stiffness decreases by 64.9% to 79.2%. The Skeletal-SP units show better compressive resistance than Skeletal-IWP units. This study provides new valuable insights for structural design and applications using TPMS structures in civil engineering.
... The deformation pattern of LS can be classifiled into bending-dominated and stretching-dominated depending on the topology of structures [5,6]. The mechanical properties of the LS can be adjusted manually by changing the meso-scale characteristics of LS to meet the requirements of the specific application [7][8][9][10][11]. Therefore, it is necessary to understand the relationship between the microstructure and the LS mechanics properties in order to obtain the optimal microstructure or morphology and thus the desired structures behaviors. ...
Lattice structures are increasingly attracting attention due to their excellent mechanical properties and broad
application prospects. However, most developed lattices feature single-step deformation or single plateau stress,
which confine its multi-task applications. Herein, a novel body-centered cubic (NBCC) with two-step deformation
based on body-centered cubic (BCC) and bionic fractal design is introduced. NBCC exhibits bi-plateau stress in
the stress–strain curves. The underlying mechanism is caused by the bending and buckling deformation of the
struts. The mechanical behaviors of NBCC are investigated by finite element simulation which verified by
experiment. Compared with traditional BCC, NBCC has improved modulus of elasticity by 88 %, yield strength by
21.2 %, and specific energy absorption by 108 % when ρ′ and α are 0.07 and 0.5, respectively. Moreover, the
elastic modulus as well as yield strength increase with geometrical ratio α. The specific energy absorption tends
to maximum at the geometric ratio α = 0.6 ~ 0.7. From the Ashby map, the proposed NBCC lattice possesses high
energy absorption, exceeding most of the existing architected materials at the same density. Furthermore, the
results show that NBCC has better isotropy of elastic modulus and tends to be more isotropic material compared
to BCC. Finally, theoretical model of two-step plateau stress is established based on hinge dissipation principle.
This work opens up new insights into the use of element replacement design to create multi-step pathways that
can be applied to design engineering structures with multiple tasks and application for impact protection
... Many design applications [17][18][19][20][21] have been limited to a small catalog of ad-hoc lattices (e.g., kagome, octet, and octahedron [22][23][24][25][26][27][28], which have been identified through a combination of intuition and trial-and-error over the years. While the catalog-based search space can be enriched by tuning geometric features 29 or base material properties 19 , it is strongly limited in topological tunability and fails to exploit the full range of achievable designs and hence of achievable effective metamaterial properties. Many truss optimization solutions have adopted heuristic search strategies to find optimal structures by iteratively adjusting the active beams and/or nodes in the design domain, according to mechanics-based criteria [30][31][32][33] . ...
The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials—truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.
... However, inclined struts are commonly seen in various strut-based lattice designs. The three main reasons are (i) to enhance the diversity of lattice topology otherwise all lattice designs will comprise horizontal and vertical struts, (ii) to help mitigate anisotropic properties [6,7] lattice topologies consisting of only horizontal and vertical struts often exhibit prominent anisotropy, and (iii) to facilitate the use of the Maxwell criterion (m) by increasing the number of struts to make > 0 (stretching-dominated) this allows control of the lattice deformation mechanism according to the Gibson-Ashby model [8]. However, from a PBF-AM perspective, dealing with inclined struts is almost unavoidable. ...
Additively manufactured metal lattices or metamaterials are often used in the form of conformal lattices. A prerequisite for realizing the design potential of these conformal lattice components is to ensure that each submillimetre to millimetre scale constituent strut achieves a uniform cross-sectional profile along its strut axis. However, in the design space of conformal lattices, the inclination angles of the constituent struts can vary considerably, even when simple cubic unit cells are used. This poses a challenge in selecting the diameter of each inclined strut for powder bed fusion (PBF) additive manufacturing (AM), because the manufacturable diameter of a strut is closely related to its inclination angle. On the other hand, while choosing a thick strut diameter could ensure a uniform cross-sectional profile for an inclined strut, this often results in an excessive lattice relative density, thereby hindering lightweight design. Through the design and detailed characterization of a complex Ti-6Al-4V conformal lattice, we show that the challenge of choosing the strut diameter versus strut inclination angle in a complex conformal lattice structure can be well answered by using a strut additive continuity model recently developed by the authors.
... Not surprising, these observations attribute to the proportion of the structures. A high β corresponds to a SC tube dominated structure where most of the materials are axially aligned in the [1 0 0] direction while a low β corresponds to an FCC plate dominated structure where stronger localization is observed in the [1 0 0] direction [33]. At a transition stage, with β = 0.48, near-isotropic structures can be realized. ...
... They are a multidisciplinary research community encompassing structural analysis, materials science, mechanics, and engineering. As a result, research groups have aimed to design functionally graded material properties of composites to achieve improved physical and mechanical behaviors that cannot be obtained from conventional engineering materials [6][7][8][9]. ...
... Optimization-based approaches are usually referred to as topology optimization, which is also known as inverse homogenization methods [21,22]. Although these techniques have been extensively utilized in prior research [23][24][25][26][27][28], they pose a challenge in 3D lattice inverse design, primarily due to their computationally intensive nature. Additionally, the structures generated by these methods may not be easily manufacturable [18,22,29]. ...
This paper investigates the feasibility of data-driven methods in automating the engineering design process, specifically studying inverse design of cellular mechanical metamaterials. Traditional methods of designing cellular materials typically rely on trial and error or iterative optimization, which often leads to limited productivity and high computational costs. While data-driven approaches have been explored for the inverse design of cellular materials, many of these methods lack robustness and fail to consider the manufacturability of the generated structures. This study aims to develop an efficient inverse design methodology that accurately generates mechanical metamaterial while ensuring the manufacturability of predicted structures. To achieve this, we have created a comprehensive dataset that spans a broad range of mechanical properties by applying rotations to cubic structures synthesized from the nine cubic symmetries of cubic materials. We then employ a physics-guided neural network (PGNN) consisting of dual neural networks: a generator network, which serves as the inverse design tool, and a forward network, which acts as a physics-guided simulator. The goal is to generate desired anisotropic stiffness components with unit-cell design parameters. The results of our inverse model are examined with three distinct datasets and demonstrate high computational efficiency and prediction accuracy compared to conventional methods.
... Strut-based lattice structures are often chosen for their simplicity of design, but strut-based topologies have also been generated from topological optimization to maximize the efficiency of material distribution within the lattice structure (Xiao et al., 2018a;Xu et al., 2016). Alternative topologies to strut-based structures are lattice structures with unit cells based on triply periodic minimal surfaces (TPMS) such as the Schoen gyroid, Schwartz diamond and Neovius (Maconachie et al., 2019). ...
Graded lattice scaffolds based on rhombic dodecahedral (RD) elementary unit cell geometry were manufactured in 316L stainless steel (SS) by laser powder bed fusion (LPBF). Two different strategies based on varying strut thickness layer-by-layer in the building direction were adopted to obtain the graded scaffolds: a) decreasing strut size from core to edge to produce the dense-in (DI) structure and b) increasing strut size in the same direction to produce the dense-out (DO) structure. Both graded structures (DI and DO) were constructed with specular symmetry with respect to the central horizontal axis. Structural, mechanical, and biological characterizations were carried out to evaluate feasibility of designing appropriate biomechanical performances of graded scaffolds in the perspective of bone tissue regeneration. Results showed that mechanical behavior is governed by graded geometry, while printing parameters influence structural properties of the material such as density, textures, and crystallographic phases. The predominant failure mechanism in graded structures initiates in correspondence of thinner struts, due to high stress concentrations on strut junctions. Biological tests evidenced better proliferation of cells in the DO graded scaffold, which in turn exhibits mechanical properties close to cortical bone. The combined control of grading strategy, printing parameters and elementary unit cell geometry can enable implementing scaffolds with improved biomechanical performances for bone tissue regeneration.
... In recent studies, the triply periodic minimal surface (TPMS) lattice structures generated by implicit functions are attractive for industrial applications owing to their superior mechanical properties [15][16][17][18][19][20]. The anisotropic property of TPMS-based lattice structure is one of the significant characteristics [21][22][23][24] and the geometric feature is the key factor affecting the anisotropy performance [25]. To further enhance the mechanical properties of components in lightweight design application, it is worth investigating the anisotropic performance of the lattice structures [26]. ...
... Up till now, inclined struts have been commonly included in lattice design (Fig. 1). Their main roles may be summarized as follows: (i) to just enhance the diversity of lattice topology, (ii) to help mitigate the anisotropic properties of some lattice designs [113,114], and (iii) to facilitate the use of the Maxwell criterion by increasing the total strut number to make m > 0 so as to control the lattice deformation mechanism by the G-A model (for l /d > 5) [10]. One such example is octet-truss lattices (m > 0), but they are much weaker than cubic lattices of the same density [61,63]. ...
The Gibson-Ashby (G-A) model has been instrumental in the design of additively manufactured (AM-ed) metal lattice materials or mechanical metamaterials. The first part of this work reviews the proposition and formulation of the G-A model and emphasizes that the G-A model is only applicable to low-density lattice materials with strut length-to-diameter ratios greater than 5. The second part evaluates the applicability of the G-A model to AM-ed metal lattice materials and reveals the fundamental disconnections between them. The third part assesses the
deformation mechanisms of AM-ed metal lattices in relation to their strut length-to-diameter ratios and identifies
that AM-ed metal lattices deform by concurrent bending, stretching, and shear, rather than just stretching or
bending considered by the G-A model. Consequently, mechanical property models coupling stretching, bending and shear deformation mechanisms are developed for various lattice materials, which show high congruence with experimental data. The last part discusses new insights obtained from these remedies into the design of strong and stiff metal lattices. In particular, we recommend that the use of inclined struts be avoided.
This study investigates the mechanical properties and fatigue performance of Ti6Al4V cellular lattice materials (CLMs) featuring five distinct unit cell types (BCC‐Z, BCC, Octet, Truncated cuboctahedron [TCO], and Trabecular) at a relative density of 25%. Compression tests were conducted to assess static properties, including Young's modulus and yield strength. Subsequently, compression–compression fatigue tests ( R = 0.1) were performed to evaluate fatigue behavior. Acoustic emission analysis was employed during static and fatigue tests to explore the potential for failure prediction. Results reveal that BCC‐Z and TCO exhibit slightly higher Young's moduli, surpassing 20 GPa, while BCC, Octet, and Trabecular display moduli ranging from 6 to 12 GPa. Regarding normalized fatigue behavior, BCC‐Z demonstrates superior fatigue resistance, followed by TCO. Notably, the acoustic emission parameters significantly correlate with the unit cell type. Lastly, a strong relationship between the initiation of failure and changes in acoustic emission parameters is observed, establishing a meaningful link between the static and fatigue curves and acoustic emission results.
The periodic truss structures in mechanical metamaterials have received widespread attention, yet stress concentration at nodes promotes the failure of truss structures, in particular under tension load. Here we report that tensile properties (in particular, ductility) of truss structures can be remarkedly improved by replacing the surface at the node with a triply periodic minimal surface (TPMS) structure. The intrinsic stress concentration at nodes in truss lattice structure can be readily dissipated at TPMS-modified nodes and transferred to the connected ligaments, leading to the enhanced tensile ductility. Moreover, the efficacy of TPMS modification is found to be correlated with the node coordination number, where lattice structures with a high coordination number of nodes possess better property improvements. The role of TPMS nodes on stress transfer and deformation de-localization behaviors is well interpreted by the in-situ microstructural analyses as well as finite element modeling simulations.
Thoughtfully engineered lattice materials offer a canvas for tailoring a diverse range of mechanical attributes. Yet, their pronounced anisotropy and inherently unstable nonlinear mechanical behavior curtail their suitability for energy absorption applications. Here, we propose the adoption of lightweight plate lattice structures for energy absorption and loading support. This involves strategically integrating plates into the rhombic dodecahedron framework, facilitating elastic isotropy and nearly yield isotropy. The proposed plate lattices showcase remarkable properties, including an exceptionally high bulk modulus that reaches the HS bound. Moreover, they boast an enhanced energy absorption capacity, surpassing that of the rhombic dodecahedron truss lattice by up to 2.58 times. The effects of relative density and loading direction on the mechanical properties were thoroughly explored through numerical simulations. Simulation predictions were rigorously validated through experimental verification. The failure modes of plate samples transition from being dependent on the loading direction, leading to collapse, to becoming stable regardless of the loading direction as the relative density increases from 0.1 to 0.2. It is noteworthy that plate lattices demonstrate SEA values comparable to those of shell lattices, even surpassing the stiffest isotropic plate lattice. These characteristics underscore their potential for applications in loading support and energy absorption.
In order to comprehensively understand the mechanical behavior of biological entities and aerospace applications subjected to hypergravity environments, we delve into the impact of hypergravity on the equivalent compliance of cubic lattice structures. Capitalizing on the periodic spatial distribution, we employ a unit cell methodology to deduce the homogenized stress-strain relationship for the lattice structures, subsequently obtaining the associated equivalent compliance. The equivalent compliance can be conveniently reduced to instances without hypergravity influence. Furthermore, numerical simulations are executed to validate the derivations and to illustrate that hypergravity indeed affects the mechanical properties of lattice structures. We introduce a non-dimensional hypergravity factor, which quantifies the impact of hypergravity magnitude relative to the Young’s modulus of the base material. Our findings reveal that the hypergravity factor influences perpendicular compliance quadratically and parallel compliance linearly. Simultaneously, the perpendicular shear compliance remains unaffected, whereas the parallel shear compliance experiences an inverse effect. Additionally, the lattice structure transforms into a gradient material oriented in the hypergravity direction, consequently generating a scale effect.
With the recent development of additive manufacturing technology represented by 3D printers, various lattice structures have emerged because their mechanical properties can be designed artificially according to their structural characteristics. This study aims to develop knowledge that will contribute to the design of lattice structures with tailored and desired mechanical properties. Among the many unit cell topologies in the lattice structures that have been designed in numerous studies, there are several basic structures and a variety of additional structures that were derived from these basic structures. However, there is a degree of ambiguity as to which of these structures should be regarded as the basic structures. In this study, the basic structures of a lattice structure with orthogonal symmetry were derived by determining the edges of the struts using a mathematical combination approach. Then, the deformation characteristics of them were evaluated through elasto-plastic finite element analysis. Five basic structures were determined by designing the unit cell of the lattice structure such that the struts pass through the vertices, the midpoints of the edges, and the centers of the faces in the cubic design region. It was found that structures with the same strut directions in their unit cells have similar stiffness distributions, and the basic structures obtained can be classified accordingly into three types. Finite element analysis of the compression of a micro-lattice structure composed of these basic structures confirmed that different deformation characteristics appeared during plastic collapse for each type of classified basic structure, and that the strut deformation type determined the overall deformation characteristics. Evaluation of the stiffness and the plastic collapse stress showed that with stretch-dominant strut deformation, lattice structures can be made lighter in weight while still maintaining some stiffness and some plastic collapse stress.
Mechanical Metamaterials (MMs) are artificially designed structures with extraordinary properties that are dependent on micro architectures and spatial tessellations of unit cells, rather than constitutive compositions. They have demonstrated promising and attractive application potentials in practical engineering. Recently, how to rationally design novel MMs and discover their multifunctional behaviors has received tremendous discussions with rapid progress, particularly in the last ten years with an enormous increase of publications and citations. Herein, we present a comprehensive overview of considerable advances of MMs, including critical focuses at different scales, forward and inverse design mechanisms with optimization formulations, micro architectures of unit cells, and their spatial tessellations in discovering novel MMs and future prospects. The implications in clarifying the four focuses at levels from the global to the physical in MMs are highlighted, that is, unique structures designed for unique functions, unique micro unit cells placed in unique locations, unique micro unit cells designed for unique properties and unique micro unit cells evaluated by unique mechanisms. We examine the inverse designs of MMs with intrinsic mechanisms of structure-property driven characteristics to achieve unprecedented behaviors, which are involved into material designs and multiscale designs. The former primarily optimizes micro architectures to explore novel MMs, and the latter focuses on micro architectures and spatial tessellations to promote multifunctional applications of MMs in engineering. Finally, we propose several promising research topics with serious challenges in design formulations, micro architectures, spatial tessellations and industrial applications.
When a cubic lattice packaged by a boundary layer is subjected to a mechanical and temperature load, the force and length change of the bonds are equivalently evaluated by the average stress and strain of the unit cell. Provided a displacement gradient variation at a certain stress state, the variation of stress related to the strain variation provides the effective stiffness of the material at the corresponding configuration. It is discovered that the effective elasticity and thermal expansion coefficient can be tailored by the prestress through the boundary layer, which generates a configurational stress. Because the bonds of a cubic lattice depend on the material types, we consider the harmonic potential of springs for cellular lattices and Hertz's contact potential of balls for granular lattices, respectively. The cubic symmetry of effective elasticity is demonstrated for the three types of cubic lattices. By taking the orientational average, isotropic elastic constants can be obtained for randomly oriented lattices. As the bond length changes with the prestress of the boundary layer and controls the thermoelastic behavior, a novel design method of lattice-based materials confined in a spherical shell is demonstrated to achieve zero thermal expansion and a positive temperature derivative of elasticity.
Low-density, high-strength, ductile metal mechanical metamaterials are in high demand for engineering applications but remain out of reach by the Gibson-Ashby model. Here we demonstrate a new design concept based on a generalized theoretical model or an extended Gibson-Ashby model to overcome this challenge. We show that the deformation mechanism of a strut-based metal lattice material depends on its strut length-to-diameter (l/d) ratio, and the key to maximizing its strength is to reduce its l/d ratio to the minimum manufacturable level without increasing the lattice density. Inspired by this design concept, we investigate three strut remodeling strategies to strengthen metal lattices. Our low-density (1.6 g/cm3) Ti-6Al-4V lattices designed according to Wolff's law of bone remodeling achieve exceptional strength (>400 MPa) compared to all cellular metallic materials of equivalent density reported so far (2-8 times their strength). Our l/d-centered design concept is expected to inspire the design of more metal metamaterials.
The stiffness and toughness of topology optimised structures containing lattices under three-point bending is studied. A porosity constraint is introduced to control the proportion of lattice generated while optimising the beams for minimum compliance. A novel tetrahedron element-based lattice with tapered trusses and a near-isotropic elastic response is developed to accurately map the relative densities from topology optimisation to a 3D printable structure. Topology optimised solid structures (0 % porosity), used for benchmarking, are found to have high stiffness but can be susceptible to buckling. At the other extreme, structures comprising entirely of lattice (100 % porosity) are shown to have low initial stiffness and low residual toughness. An experimental parametric study reveals that porosity can be tailored between these two bounds to achieve both high stiffness and high residual toughness.
The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials -- truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized properties, including exceptionally stiff, auxetic, and pentamode-like designs. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.
Ceramics have some of the highest strength- and stiffness-to-weight ratios of any material but are suboptimal for use as structural
materials because of their brittleness and sensitivity to flaws. We demonstrate the creation of structural metamaterials composed
of nanoscale ceramics that are simultaneously ultralight, strong, and energy-absorbing and can recover their original shape
after compressions in excess of 50% strain. Hollow-tube alumina nanolattices were fabricated using two-photon lithography,
atomic layer deposition, and oxygen plasma etching. Structures were made with wall thicknesses of 5 to 60 nanometers and densities
of 6.3 to 258 kilograms per cubic meter. Compression experiments revealed that optimizing the wall thickness-to-radius ratio
of the tubes can suppress brittle fracture in the constituent solid in favor of elastic shell buckling, resulting in ductile-like
deformation and recoverability.
Additive manufacturing, otherwise known as three-dimensional (3D) printing, is driving major innovations in many areas, such as engineering, manufacturing, art, education and medicine. Recent advances have enabled 3D printing of biocompatible materials, cells and supporting components into complex 3D functional living tissues. 3D bioprinting is being applied to regenerative medicine to address the need for tissues and organs suitable for transplantation. Compared with non-biological printing, 3D bioprinting involves additional complexities, such as the choice of materials, cell types, growth and differentiation factors, and technical challenges related to the sensitivities of living cells and the construction of tissues. Addressing these complexities requires the integration of technologies from the fields of engineering, biomaterials science, cell biology, physics and medicine. 3D bioprinting has already been used for the generation and transplantation of several tissues, including multilayered skin, bone, vascular grafts, tracheal splints, heart tissue and cartilaginous structures. Other applications include developing high-throughput 3D-bioprinted tissue models for research, drug discovery and toxicology.
T cells that accompany allogeneic hematopoietic grafts for treating leukemia enhance engraftment and mediate the graft-versus-leukemia
effect. Unfortunately, alloreactive T cells also cause graft-versus-host disease (GVHD). T cell depletion prevents GVHD but
increases the risk of graft rejection and leukemic relapse. In human transplants, we show that donor-versus-recipient natural
killer (NK)–cell alloreactivity could eliminate leukemia relapse and graft rejection and protect patients against GVHD. In
mice, the pretransplant infusion of alloreactive NK cells obviated the need for high-intensity conditioning and reduced GVHD.
NK cell alloreactivity may thus provide a powerful tool for enhancing the efficacy and safety of allogeneic hematopoietic
transplantation.
Additive manufacturing (AM) technology has been researched and developed for more than 20 years. Rather than removing materials, AM processes make three-dimensional parts directly from CAD models by adding materials layer by layer, offering the beneficial ability to build parts with geometric and material complexities that could not be produced by subtractive manufacturing processes. Through intensive research over the past two decades, significant progress has been made in the development and commercialization of new and innovative AM processes, as well as numerous practical applications in aerospace, automotive, biomedical, energy and other fields. This paper reviews the main processes, materials and applications of the current AM technology and presents future research needs for this technology.
Tissue engineering scaffold is a biological substitute that aims to restore, to maintain, or to improve tissue functions. Currently available manufacturing technology, that is, additive manufacturing is essentially applied to fabricate the scaffold according to the predefined computer aided design (CAD) model. To develop scaffold CAD libraries, the polyhedrons could be used in the scaffold libraries development. In this present study, one hundred and nineteen polyhedron models were evaluated according to the established criteria. The proposed criteria included considerations on geometry, manufacturing feasibility, and mechanical strength of these polyhedrons. CAD and finite element (FE) method were employed as tools in evaluation. The result of evaluation revealed that the close-cellular scaffold included truncated octahedron, rhombicuboctahedron, and rhombitruncated
cuboctahedron. In addition, the suitable polyhedrons for using as open-cellular scaffold libraries included hexahedron, truncated octahedron, truncated hexahedron,
cuboctahedron, rhombicuboctahedron, and rhombitruncated cuboctahedron. However, not all pore size to beam thickness ratios (PO : BT) were good for making the open-cellular scaffold. The PO : BT ratio of each library, generating the enclosed pore inside the scaffold, was excluded to avoid the impossibility of material removal after the fabrication. The close-cellular libraries presented the constant porosity
which is irrespective to the different pore sizes. The relationship between PO : BT ratio and porosity of open-cellular scaffold libraries was displayed in the form of Logistic Power function. The possibility of merging two different types of libraries to produce the composite structure was geometrically evaluated in terms of the intersection index and
was mechanically evaluated by means of FE analysis to observe the stress level. The couples of polyhedrons presenting low intersection index and high stress level were excluded. Good couples for producing the reinforced scaffold were
hexahedron-truncated hexahedron and cuboctahedron-rhombitruncated cuboctahedron.
Ultralight (<10 milligrams per cubic centimeter) cellular materials are desirable for thermal insulation; battery electrodes; catalyst supports; and acoustic, vibration, or shock energy damping. We present ultralight materials based on periodic hollow-tube microlattices. These materials are fabricated by starting with a template formed by self-propagating photopolymer waveguide prototyping, coating the template by electroless nickel plating, and subsequently etching away the template. The resulting metallic microlattices exhibit densities ρ ≥ 0.9 milligram per cubic centimeter, complete recovery after compression exceeding 50% strain, and energy absorption similar to elastomers. Young's modulus E scales with density as E ~ ρ(2), in contrast to the E ~ ρ(3) scaling observed for ultralight aerogels and carbon nanotube foams with stochastic architecture. We attribute these properties to structural hierarchy at the nanometer, micrometer, and millimeter scales.
In this paper, we examine prospects for the manufacture of patient-specific biomedical implants replacing hard tissues (bone), particularly knee and hip stems and large bone (femoral) intramedullary rods, using additive manufacturing (AM) by electron beam melting (EBM). Of particular interest is the fabrication of complex functional (biocompatible) mesh arrays. Mesh elements or unit cells can be divided into different regions in order to use different cell designs in different areas of the component to produce various or continually varying (functionally graded) mesh densities. Numerous design elements have been used to fabricate prototypes by AM using EBM of Ti-6Al-4V powders, where the densities have been compared with the elastic (Young) moduli determined by resonant frequency and damping analysis. Density optimization at the bone-implant interface can allow for bone ingrowth and cementless implant components. Computerized tomography (CT) scans of metal (aluminium alloy) foam have also allowed for the building of Ti-6Al-4V foams by embedding the digital-layered scans in computer-aided design or software models for EBM. Variations in mesh complexity and especially strut (or truss) dimensions alter the cooling and solidification rate, which alters the alpha-phase (hexagonal close-packed) microstructure by creating mixtures of alpha/alpha' (martensite) observed by optical and electron metallography. Microindentation hardness measurements are characteristic of these microstructures and microstructure mixtures (alpha/alpha') and sizes.
Periodic cellular metals with honeycomb and corrugated topologies are widely used for the cores of light weight sandwich panel structures. Honeycombs have closed cell pores and are well suited for thermal protection while also providing efficient load support. Corrugated core structures provide less efficient and highly anisotropic load support, but enable cross flow heat exchange opportunities because their pores are continuous in one direction. Recent advances in topology design and fabrication have led to the emergence of lattice truss structures with open cell structures. These three classes of periodic cellular metals can now be fabricated from a wide variety of structural alloys. Many topologies are found to provide adequate stiffness and strength for structural load support when configured as the cores of sandwich panels. Sandwich panels with core relative densities of 2-10% and cell sizes in the millimetre range are being assessed for use as multifunctional structures. The open, three-dimensional interconnected pore networks of lattice truss topologies provide opportunities for simultaneously supporting high stresses while also enabling cross flow heat exchange. These highly compressible structures also provide opportunities for the mitigation of high intensity dynamic loads created by impacts and shock waves in air or water. By filling the voids with polymers and hard ceramics, these structures have also been found to offer significant resistance to penetration by projectiles.
The rigidity of a network of elastic beams crucially depends on the specific
details of its structure. We show both numerically and theoretically that there
is a class of isotropic networks which are stiffer than any other isotropic
network with same density. The elastic moduli of these \textit{stiffest elastic
networks} are explicitly given. They constitute upper-bounds which compete or
improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of
criteria (necessary and sufficient conditions) to identify these networks, and
show that their displacement field under uniform loading conditions is affine
down to the microscopic scale. Finally, examples of such networks with periodic
arrangement are presented, in both two and three dimensions.
This first comprehensive overview of the modern aspects of biomineralization represents life and materials science at its best: Bioinspired pathways are the hot topics in many disciplines and this holds especially true for biomineralization. Here, the editors -- all well-known members of associations and prestigious institutes -- have assembled an international team of renowned authors to provide first-hand research results. From the contents: VOLUME 1: BIOLOGICAL ASPECTS AND STRUCTURE FORMATION. Silica-Hydrated Polysilicondioxide. Iron Sulfides and Oxides. Calcium Carbonates and Sulfates. Calcium Phosphates. VOLUME 2: BIOMIMETIC AND BIOINSPIRED CHEMISTRY. Biometic Model Systems in Biomineralization. The Biomineral Approach to Bionics. Bio-inspired Materials Synthesis. Bio-supported Materials Chemistry. Protein Cages as Size-contrained Reaction Vessels. Encapsulation. Imaging of Internal Nanostructures of Biominerals. VOLUME 3: MEDICAL AND CLINICAL ASPECTS. Bone. Teeth. Pathological Calcifications. An interdisciplinary must-have account, for biochemists, bioinorganic chemists, lecturers in chemistry and biochemistry, materials scientists, biologists, and solid state physicists. With forewords by Jeremy D. Pickett-Heaps, Stephen Mann, and Wolfgang Pompe.
Most AM processes require post-processing after part building to prepare the part for its intended form, fit and/or function. Depending upon the AM technique, the reason for post-processing varies. For purposes of simplicity, this chapter will focus on post-processing techniques which are used to enhance components or overcome AM limitations. These include:
Preface to the first edition Preface to the second edition 1. Materials: structure, properties and performance 2. Elasticity and viscoelasticity 3. Plasticity 4. Imperfections: point and line defects 5. Imperfections: interfacial and volumetric defects 6. Geometry of deformation and work-hardening 7. Fracture: macroscopic aspects 8. Fracture: microscopic aspects 9. Fracture testing 10. Solid solution, precipitation and dispersion strengthening 11. Martensitic transformation 12. Speciality materials: intermetallics and foams 13. Creep and superplasticity 14. Fatigue 15. Composite materials 16. Environmental effects.
Cellular solids include engineering honeycombs and foams (which can now be made from polymers, metals, ceramics, and composites) as well as natural materials, such as wood, cork, and cancellous bone. This new edition of a classic work details current understanding of the structure and mechanical behavior of cellular materials, and the ways in which they can be exploited in engineering design. Gibson and Ashby have brought the book completely up to date, including new work on processing of metallic and ceramic foams and on the mechanical, electrical and acoustic properties of cellular solids. Data for commercially available foams are presented on material property charts; two new case studies show how the charts are used for selection of foams in engineering design. Over 150 references appearing in the literature since the publication of the first edition are cited. It will be of interest to graduate students and researchers in materials science and engineering.
There has been considerable interest in materials exhibiting negative or zero compressibility. Such
materials are desirable for various applications. A number of models or mechanisms have been proposed
to characterize the unusual phenomena of negative linear compressibility (NLC) and negative area
compressibility (NAC) in natural or synthetic systems. In this paper we propose a general design technique
for finding metamaterials with negative or zero compressibility by using a topology optimization
approach. Based on the bi-directional evolutionary structural optimization (BESO) method, we establish a
systematic computational procedure and present a series of designs of orthotropic materials with various
magnitudes of negative compressibility, or with zero compressibility, in one or two directions. A physical
prototype of one of such metamaterials is fabricated using a 3D printer and tested in the laboratory under
either unidirectional loading or triaxial compression. The experimental results compare well with the
numerical predictions. This research has demonstrated the feasibility of designing and fabricating
metamaterials with negative or zero compressibility and paved the way towards their practical
applications.
Treatment of large segmental bone defects, especially in load bearing areas, is a complex procedure in orthopedic surgery. The usage of additive manufacturing processes enables the creation of customized bone implants with arbitrary open-porous structure satisfying both the mechanical and the biological requirements for a sufficient bone ingrowth. Aim of the present numerical study was to optimize the geometrical parameters of open-porous titanium scaffolds to match the elastic properties of human cortical bone with respect to an adequate pore size. Three different scaffold designs (cubic, diagonal and pyramidal) were numerically investigated by using an optimization approach. Beam elements were used to create the lattice structures of the scaffolds. The design parameters strut diameter and pore size ranged from 0.2 to 1.5 mm and from 0 to 3.0 mm, respectively. In a first optimization step, the geometrical parameters were varied under uniaxial compression to obtain a structural modulus of 15 GPa (Young׳s modulus of cortical bone) and a pore size of 800 µm was aimed to enable cell ingrowth. Furthermore, the mechanical behavior of the optimized structures under bending and torsion was investigated. Results for bending modulus were between 9.0 and 14.5 GPa. In contrast, shear modulus was lowest for cubic and pyramidal design of approximately 1 GPa. Here, the diagonal design revealed a modulus of nearly 20 GPa. In a second step, large-sized bone scaffolds were created and placed in a biomechanical loading situation within a 30 mm segmental femoral defect, stabilized with an osteosynthesis plate and loaded with physiological muscle forces. Strut diameter for the 17 sections of each scaffold was optimized independently in order to match the biomechanical stability of intact bone. For each design, highest strut diameter was found at the dorsal/medial site of the defect and smallest strut diameter in the center. In conclusion, we demonstrated the possibility of providing optimized open-porous scaffolds for bone regeneration by considering both mechanical and biological aspects. Furthermore, the results revealed the need of the investigation and comparison of different load scenarios (compression, bending and torsion) as well as complex biomechanical loading for a profound characterization of different scaffold designs. The usage of a numerical optimization process was proven to be a feasible tool to reduce the amount of the required titanium material without influencing the biomechanical performance of the scaffold negatively. By using fully parameterized models, the optimization approach is adaptable to other scaffold designs and bone defect situations.
Cellular structures with highly controlled micro-architectures are promising materials for orthopedic applications that require bone-substituting biomaterials or implants. The availability of additive manufacturing techniques has enabled manufacturing of biomaterials made of one or multiple types of unit cells. The diamond lattice unit cell is one of the relatively new types of unit cells that are used in manufacturing of regular porous biomaterials. As opposed to many other types of unit cells, there is currently no analytical solution that could be used for prediction of the mechanical properties of cellular structures made of the diamond lattice unit cells. In this paper, we present new analytical solutions and closed-form relationships for predicting the elastic modulus, Poisson's ratio, critical buckling load, and yield (plateau) stress of cellular structures made of the diamond lattice unit cell. The mechanical properties predicted using the analytical solutions are compared with those obtained using finite element models. A number of solid and porous titanium (Ti6Al4V) specimens were manufactured using selective laser melting. A series of experiments were then performed to determine the mechanical properties of the matrix material and cellular structures. The experimentally measured mechanical properties were compared with those obtained using analytical solutions and finite element (FE) models. It has been shown that, for small apparent density values, the mechanical properties obtained using analytical and numerical solutions are in agreement with each other and with experimental observations. The properties estimated using an analytical solution based on the Euler-Bernoulli theory markedly deviated from experimental results for large apparent density values. The mechanical properties estimated using FE models and another analytical solution based on the Timoshenko beam theory better matched the experimental observations.
Advances introduced by additive manufacturing have significantly improved the ability to tailor scaffolds architecture, enhancing the control over microstructural features. This has led to a growing interest in the development of innovative scaffold designs, as testified by the increasing amount of research activities devoted to the understanding of the correlation between scaffold topological features and its resulting properties, in order to find architectures capable to optimally trade-off between often conflicting requirements (such as the biological and the mechanical ones). Main aim of this article is to provide a review and propose a classification of existing methodologies for scaffold design and optimization in order to address key issues and help in deciphering the complex link between design criteria and resulting scaffold properties.
An orthotropic material is characterized by nine independent moduli. The ratios between the Young’s moduli in three directions are indicative of the level of orthotropy and the bulk modulus is indicative of the overall stiffness. In this paper we propose a method for designing the stiffest orthotropic material which has prescribed ratios for Young’s moduli. The material is modeled as a microstructure in a periodic unit cell. By using the homogenization method, the elasticity tensors are calculated and its compliance matrix is derived. A Lagrangian function is constructed to combine the objective and multiple equality constraints. To enable a bi-section search algorithm, the upper and lower bounds on those multipliers are derived by using a strain energy approach. The overall optimization is based on the bi-directional evolutionary structural optimization (BESO) method. Examples of various orthotropy ratios are investigated. The topology presents a constant pattern of material re-distributed along the strongest axis while the overall stiffness is maintained.
Rapid prototyping allows titanium porous parts with mechanical properties close to that of bone tissue to be obtained. In this article, porous parts of the Ti-6Al-4V alloy with three levels of porosity were obtained by selective laser melting with two different energy inputs. Thermal treatments were performed to determine the influence of the microstructure on the mechanical properties. The porous parts were characterized by both optical and scanning electron microscopy. The effective modulus, yield and ultimate compressive strength were determined by compressive tests. The martensitic α' microstructure was observed in all of the as-processed parts. The struts resulting from the processing conditions investigated were thinner than those defined by CAD models, and consequently, larger pores and a higher experimental porosity were achieved. The use of the high-energy input parameters produced parts with higher oxygen and nitrogen content, their struts that were even thinner and contained a homogeneous porosity distribution. Greater mechanical properties for a given relative density were obtained using the high-energy input parameters. The as-quenched martensitic parts showed yield and ultimate compressive strengths similar to the as-processed parts, and these were greater than those observed for the fully annealed samples that had the lamellar microstructure of the equilibrium α+β phases. The effective modulus was not significantly influenced by the thermal treatments. A comparison between these results and those of porous parts with similar geometry obtained by selective electron beam melting shows that the use of a laser allows parts with higher mechanical properties for a given relative density to be obtained.
Resonant ultrasound spectroscopy (RUS) allows to accurately characterize the complete set of elastic constants of an anisotropic material from a set of measured mechanical resonant frequencies of a specimen. This method does not suffer from the drawbacks and limitations of the conventional sound velocity approach, but has been reported to fail to measure bone because of its strong viscoelastic damping. In this study, we take advantage of recent developments of RUS to overcome this limitation. The frequency response of a human cortical bone specimen (about 5×7×7mm(3)) was measured between 100 and 280kHz. Despite an important overlapping of the resonant peaks 20 resonant frequencies could be retrieved by using a dedicated signal processing method. The experimental frequencies were progressively matched to the frequencies predicted by a model of the sample whose elastic constants were adjusted. The determined diagonal elastic constants were in good agreement with concurrent sound velocity measurements performed in the principal directions of the specimen. This study demonstrates that RUS is suitable for an accurate measurement of cortical bone anisotropic elasticity. In particular, precision of measured Young and shear moduli is about 0.5%.
The linking of computational design with precision solid freeform fabrication has tremendous potential for producing tissue scaffolds with tailored properties. We consider a new approach to optimizing the architecture of scaffolds based on jointly maximizing scaffold stiffness and diffusive transport in the interconnected pores. The stiffness of the scaffolds is matched to that of bone by choosing a suitable scaffold porosity. Moreover, the templates can be scaled to achieve target pore sizes whilst preserving their elastic and diffusive properties. The resultant structures have two major design benefits. First, the scaffolds do not have directions of low stiffness. In contrast, the Young's modulus of conventional layered-grid designs can be 86% less under diagonally-aligned loads than under axis-aligned loads. Second, the mass of the scaffold is used efficiently throughout the structure rather than being clumped in non load-bearing regions. We fabricate prototypes of the implants using selective laser melting and test their elastic properties. Excellent agreement between theory and experiment provides important confirmation of the viability of this route to scaffold design and fabrication.
In this paper, a 3D macro/micro finite element analysis (FEA) modeling approach and a 3D macro/micro analytical modeling approach are proposed for predicting the failure strengths of 3D orthogonal woven CFRP composites. These approaches include two different scale levels, macro- and micro-level. At the macro-level, a relatively coarse structural model is used to study the overall response of the structure. At the micro-level, the laminate block microstructure is modeled in detail for investigating the failure mechanisms of 3D orthogonal woven CFRP composites. The FEA and analytical models developed previously [Tan P, Tong L, Steven GP. Modeling approaches for 3D orthogonal woven composites, Journal of Reinforced Plastics and Composites, 1998:17;545–577] are employed to predict the mechanical properties of 3D orthogonal woven CFRP composites. All models presented in this paper are validated by comparing the relevant predictions with the experimental results, which were reported earlier in Part I of the paper [Tan P, Tong L, Steven GP. Behavior of 3D orthogonal woven CFRP composites. Part I. Experimental investigation, Composites, Part A: Applied Science and Manufacturing, 2000:31;259–71]. The comparison shows that there is a good agreement for the mechanical properties. An acceptable agreement exists for the failure strength in the x or stuffer yarn direction even though the FEA model gives a lower bound and the analytical model gives an upper bound. However, for the failure strength in the y or filler yarn direction, the difference between the predicted and experimental results is significant due to primarily ignoring of the waviness of filler yarn in the models. A curved beam model, which considers the waviness of the filler yarn, will be presented in Part III of the paper.
This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.
Selective electron beam melting (SEBM) was successfully used to fabricate novel cellular Ti–6Al–4V structures for orthopaedic applications. Micro computer tomography (μCT) analysis demonstrated the capability to fabricate three-dimensional structures with an interconnected porosity and pore sizes suitable for tissue ingrowth and vascularization. Mechanical properties, such as compressive strength and elastic modulus, of the tested structures were similar to those of human bone. Thus, stress-shielding effects after implantation might be avoided due to a reduced stiffness mismatch between implant and bone. A chemical surface modification using HCl and NaOH induced apatite formation during in vitro bioactivity tests in simulated body fluid under dynamic conditions. The modified bioactive surface is expected to enhance the fixation of the implant in the surrounding bone as well as to improve its long-term stability.
We present a basic analysis that establishes the metrics affecting the energy absorbed by multilayer cellular media during irreversible compaction on either a mass or volume basis. The behaviors at low and high impulse levels are distinguished through the energy dissipated in the shock. The overall mass of an energy absorbing system (comprising a cellular medium and a buffer) is minimized by maximizing the non-dimensional dissipation per unit mass parameter for the cellular medium, Λ≡Umρs/σY, where Um is the dissipation per unit mass of the cellular medium, ascertained from the area under the quasi-static compressive stress/strain curve, σY the yield strength of the constituent material and ρs the density of the material used in the medium. Plots of Λ against the non-dimensional stress transmitted through the medium, σtr/σY demonstrate the relative energy absorbing characteristics of foams and prismatic media, such as honeycombs. Comparisons with these benchmark systems are used to demonstrate the superior performance of micro-lattices, especially those with hollow truss members. Numerical calculations demonstrate the relative densities and geometric configurations wherein the lattices offer benefit. Experimental results obtained for a Ni micro-lattice with hollow members not only affirm the benefits, but also demonstrate energy absorption levels substantially exceeding those predicted by analysis. This assessment highlights the new opportunities that tailored micro-lattices provide for unprecedented levels of energy absorption for protection from impulsive loads.
Popularization of portable electronics and electric vehicles worldwide stimulates the development of energy storage devices, such as batteries and supercapacitors, toward higher power density and energy density, which significantly depends upon the advancement of new materials used in these devices. Moreover, energy storage materials play a key role in efficient, clean, and versatile use of energy, and are crucial for the exploitation of renewable energy. Therefore, energy storage materials cover a wide range of materials and have been receiving intensive attention from research and development to industrialization. In this Review, firstly a general introduction is given to several typical energy storage systems, including thermal, mechanical, electromagnetic, hydrogen, and electrochemical energy storage. Then the current status of high-performance hydrogen storage materials for on-board applications and electrochemical energy storage materials for lithium-ion batteries and supercapacitors is introduced in detail. The strategies for developing these advanced energy storage materials, including nanostructuring, nano-/microcombination, hybridization, pore-structure control, configuration design, surface modification, and composition optimization, are discussed. Finally, the future trends and prospects in the development of advanced energy storage materials are highlighted.
Patient specific porous implants for the reconstruction of craniofacial defects have gained importance due to their better performance over their generic counterparts. The recent introduction of electron beam melting (EBM) for the processing of titanium has led to a one step fabrication of porous custom titanium implants with controlled porosity to meet the requirements of the anatomy and functions at the region of implantation. This paper discusses an image based micro-structural analysis and the mechanical characterization of porous Ti6Al4V structures fabricated using the EBM rapid manufacturing process. SEM studies have indicated the complete melting of the powder material with no evidence of poor inter-layer bonding. Micro-CT scan analysis of the samples indicate well formed titanium struts and fully interconnected pores with porosities varying from 49.75%-70.32%. Compression tests of the samples showed effective stiffness values ranging from 0.57(+/-0.05)-2.92(+/-0.17)GPa and compressive strength values of 7.28(+/-0.93)-163.02(+/-11.98)MPa. For nearly the same porosity values of 49.75% and 50.75%, with a variation in only the strut thickness in the sample sets, the compressive stiffness and strength decreased significantly from 2.92 GPa to 0.57 GPa (80.5% reduction) and 163.02 MPa to 7.28 MPa (93.54 % reduction) respectively. The grain density of the fabricated Ti6Al4V structures was found to be 4.423 g/cm(3) equivalent to that of dense Ti6Al4V parts fabricated using conventional methods. In conclusion, from a mechanical strength viewpoint, we have found that the porous structures produced by the electron beam melting process present a promising rapid manufacturing process for the direct fabrication of customized titanium implants for enabling personalized medicine.
Musculoskeletal tissue, bone and cartilage are under extensive investigation in tissue engineering research. A number of biodegradable and bioresorbable materials, as well as scaffold designs, have been experimentally and/or clinically studied. Ideally, a scaffold should have the following characteristics: (i) three-dimensional and highly porous with an interconnected pore network for cell growth and flow transport of nutrients and metabolic waste; (ii) biocompatible and bioresorbable with a controllable degradation and resorption rate to match cell/tissue growth in vitro and/or in vivo; (iii) suitable surface chemistry for cell attachment, proliferation, and differentiation and (iv) mechanical properties to match those of the tissues at the site of implantation. This paper reviews research on the tissue engineering of bone and cartilage from the polymeric scaffold point of view.
Porosity and pore size of biomaterial scaffolds play a critical role in bone formation in vitro and in vivo. This review explores the state of knowledge regarding the relationship between porosity and pore size of biomaterials used for bone regeneration. The effect of these morphological features on osteogenesis in vitro and in vivo, as well as relationships to mechanical properties of the scaffolds, are addressed. In vitro, lower porosity stimulates osteogenesis by suppressing cell proliferation and forcing cell aggregation. In contrast, in vivo, higher porosity and pore size result in greater bone ingrowth, a conclusion that is supported by the absence of reports that show enhanced osteogenic outcomes for scaffolds with low void volumes. However, this trend results in diminished mechanical properties, thereby setting an upper functional limit for pore size and porosity. Thus, a balance must be reached depending on the repair, rate of remodeling and rate of degradation of the scaffold material. Based on early studies, the minimum requirement for pore size is considered to be approximately 100 microm due to cell size, migration requirements and transport. However, pore sizes >300 microm are recommended, due to enhanced new bone formation and the formation of capillaries. Because of vascularization, pore size has been shown to affect the progression of osteogenesis. Small pores favored hypoxic conditions and induced osteochondral formation before osteogenesis, while large pores, that are well-vascularized, lead to direct osteogenesis (without preceding cartilage formation). Gradients in pore sizes are recommended for future studies focused on the formation of multiple tissues and tissue interfaces. New fabrication techniques, such as solid-free form fabrication, can potentially be used to generate scaffolds with morphological and mechanical properties more selectively designed to meet the specificity of bone-repair needs.
Jan 2009
M A Meyers
K K Chawla
M.A. Meyers, K.K. Chawla, Mechanical Behavior of Materials, Cambridge University
Press, Cambridge, 2009.
Jan 2011
962-965
T A Schaedler
A J Jacobsen
A Torrents
A E Sorensen
J Lian
J R Greer
L Valdevit
W B Carter
T.A. Schaedler, A.J. Jacobsen, A. Torrents, A.E. Sorensen, J. Lian, J.R. Greer, L. Valdevit,
W.B. Carter, Ultralight metallic microlattices, Science 334 (2011) 962–965.