Article

Design of lattice structures with controlled anisotropy

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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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... The indices of the elastic constants are compatible with the Voigt notation of the stiffness tensor. To calculate the elastic constants, FEA simulations were performed with the ANSYS Workbench 18.0 code, using a numerical technique that provides close matching between theoretical and experimental results of anisotropic structures [432]. Both structures were meshed with 10-node tetrahedral elements. ...
... Specifically, for ρ F S = 0.02, the stiffness map has a star-like shape, whereas forρ F S = 0.2 the stiffness map is more uniform. The variation of the anisotropy of the FS hyper unit cell and the invariance of the anisotropy of the OT unit cell can be quantified in terms of the anisotropy Zenner ratio A [432], which for cubic symmetry is defined by ...
... The present analysis shows that intertwined structures are characterized by high stiffness and high strain energy density at both microscale and macroscale, using either brittle or ductile materials for their fabrication. In addition, the intriguing concept of variable anisotropy [432] enables the relative density to be used as a design parameter to control the directional stiffness of the structure. However, this was accomplished by merging different unit cells [432], not by intertwining select beam members. ...
Thesis
Architected materials are considered the state of the art of engineering ingenuity. Specifically, mechanical metamaterials have been accentuated due to their unconventional and augmented responses. They have been gingerly investigated under the context of ultralight-ultrastiff structures for aerospace applications, tailored buckling mechanisms for energy storage, soft robotics and controlled wave propagation and designed anisotropy for tissue engineering. Albeit the plethora of remarkable results promulgating this subject, the analysis of architected materials has many questions that need to be addressed. There is no rigorous explanation for the selection of specific 3D designs that have been thoroughly utilized in the literature (regarding the selection of specific design variables and cost functions). Consequently, in practice specific structures are repeatedly used, without any explanation whether further search of the design space could not provide a substantially improved result. Therefore, the lack of understanding of the design space and the inherent physical phenomena has not elucidated the tools to obtain a globally optimal design. Thus, tailoring mechanical metamaterials is extremely arduous and has led to an obstacle in the progress of this field. This thesis aims to provide an analysis for the design of architected materials by illuminating the physical mechanisms and how to model and optimize such problems. The structure of this thesis is comprised of two main themes. The first method aims to control the mechanical performance through interconnected beam members that enhance the densification of the structure and impede catastrophic failure. The second method is related to geometrical defects that dictate the localized failure and anisotropic behavior. Furthermore, the optimization of specific design examples will be presented, employing low computational power for large design spaces and demonstrate how such design problems can be addressed, setting the framework for the systematic design and characterization of architected materials.
... In addition to porosity, mechanical properties are also affected by loading direction due to anisotropy induced by geometrical features or manufacturing method. In this study, degree of anisotropy was visualized by 3D surface plots of Young's modulus (Chen et al., 2019;Xu et al., 2016). Generally, isotropic properties result in smooth rounded plots while increased anisotropy results in sharp protrusions or depressions, like what is typically found for many traditional strut-based structures (Chen et al., 2019;Li et al., 2018;Xu et al., 2016;Kang et al., 2020). ...
... In this study, degree of anisotropy was visualized by 3D surface plots of Young's modulus (Chen et al., 2019;Xu et al., 2016). Generally, isotropic properties result in smooth rounded plots while increased anisotropy results in sharp protrusions or depressions, like what is typically found for many traditional strut-based structures (Chen et al., 2019;Li et al., 2018;Xu et al., 2016;Kang et al., 2020). We observed plots with relatively smooth and round distributions, agreeing with other existing reports for TPMS structures (Chen et al., 2019;Li et al., 2018). ...
... In this study, a comparison was made between the measured mechanical properties and those predicted by the homogenization method, an approach which has been widely used for other periodic structures (Nguyen et al., 2019;Chen et al., 2019;Arabnejad and Pasini, 2013;Li et al., 2018;Str ö mberg, 2020;Xu et al., 2016;Lu et al., 2019). We found the difference to be between 2% and 12% for the z-direction and between 27% and 62% for the xy-direction. ...
Article
Additively manufactured structures designed from triply periodic minimal surfaces (TPMSs) have been receiving attention for their potential uses in the medical, aerospace, and automobile industries. Understanding how these complex geometries can be designed to achieve particular architectural and mechanical properties is essential for tuning their function to certain applications. In this study, we created design tools for visualizing the interplay between TPMS design parameters and resulting architecture and aimed to validate a model of the relationship between structure architecture and Young's modulus. A custom MATLAB script was written to analyze structural properties for families of Schoen gyroid and Schwarz diamond structures, and a numerical homogenization scheme was performed to predict the effective Young's moduli of the structures based on their architecture. Our modeling methods were validated experimentally with polyetheretherketone (PEEK) structures created using material extrusion additive manufacturing. The architectural characteristics of the structures were determined using micro-computed tomography, and compression testing was performed to determine yield strength and Young's modulus. Two different initial build orientations were tested to determine the behavior both perpendicular and parallel to the layer deposition direction (referred to as z-direction and xy-direction, respectively). The z-direction Young's modulus ranged from 289.7 to 557.5 MPa and yield strength ranged from 10.12 to 20.3 MPa. For the xy-direction, Young's modulus ranged from 133.8 to 416.4 MPa and yield strength ranged from 3.8 to 12.2 MPa. For each initial build orientation, the mechanical properties were found to decrease with increasing porosity, and failure occurred due to both strut bending and interlayer debonding. The mechanical properties predicted by the modeling agreed with the values found for z-direction samples (difference 2-11%) but less so for xy-direction samples (difference 27-62%) due to weak interlayer bonding and print path irregularities. Ultimately, the findings presented here provide better understanding of the range of properties achievable for additive manufacturing of PEEK and encouraging results for a TPMS architecture-property model.
... [5][6][7] . 然而, 目前多数点阵结构的力学性能 呈现明显的各向异性, 它们仅在部分方向上具有较 高的承载能力 [8] . 结构的各向异性在许多应用中被 视为一种有害的特性, 特别是当各向异性结构被用 作结构构件或者能量吸收材料时, 由于工程中的载 荷往往存在不确定性, 在这种情况下自然不希望结 构部件在部分方向上的力学性能明显较弱 [9][10] . ...
... 设计或各向异性控制展开 [8][9][10][25][26][27][28][29][30][31][32] ...
... +A cos (4θ − 4ϕ) +B cos (4θ+2ϕ) + B cos (4θ − 2ϕ) +C cos (4θ) +D cos (2θ+4ϕ) + D cos (2θ − 4ϕ) + Ecos (2θ+2ϕ) +E cos (2θ − 2ϕ) + F cos (2θ) +G cos (2ϕ) +H cos (4ϕ) +I] −1(8) 式中, 参数A, B, C, D, E, F, G, H和I是常数, ...
Article
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The rise of additive manufacturing technology stimulates researchers' enthusiasm for structural innovative design. However, the anisotropy of additive manufactured materials poses certain difficulties for the prediction and design of structural mechanical properties. To accurately predict the elastic properties of polylactic acid (PLA) materials and lattice structures made in fused filament fabrication and realize the elastically isotropic design of lattice structures, firstly, this paper adopts an orthogonal anisotropic elastic model to describe the elastic behavior of the PLA materials, and obtains the nine independent elastic constants needed for the orthogonal anisotropic model through experiments and calculations. Then, a 2D compound truss lattice structure with tunable mechanical properties is designed, and its analytical expressions for the in-plane effective elastic properties and elastic isotropy condition are derived based on the representative volume element (RVE) method without considering the material anisotropy. Finally, the elastic modulus and thickness of the struts in the lattice structure are adjusted according to the anisotropy of the PLA material, and the analytical expressions of the in-plane elastic properties and elastic isotropy condition of the lattice structur are derived based on the RVE method. The results show that the orthogonal anisotropic elastic model is suitable for describing the elastic behavior of fused filament fabricated PLA materials, and the elastic modulus of PLA materials in arbitrary direction can be accurately predicted based on this model. The anisotropy of the material needs to be fully considered when predicting and designing the mechanical properties of the fused filament fabricated lattice structures. After considering the material anisotropy, the elastic isotropy design of part of lattice structures can be realized by adjusting their geometric sizes based on the RVE method.(增材制造技术的兴起激发了国内外学者对结构创新设计的热情。然而,增材制造材料的各向异性为结构力学性能的预测与设计带来了一定的困难。为了准确预测熔丝制造聚乳酸(PLA)材料和点阵结构的弹性性能,并实现点阵结构的弹性各向同性设计,首先,本文采用正交各向异性弹性模型来描述PLA材料的弹性行为,通过实验和计算得到了正交各向异性模型需要的9个独立的弹性常数。然后,设计了一种力学性能可调的二维组合桁架点阵结构,基于代表体元法,在不考虑材料各向异性的情况下推导出了其平面内等效弹性性能的解析表达式及弹性各向同性条件。最后,根据PLA材料的各向异性调整点阵结构内部杆件的弹性模量和厚度,并基于代表体元法重新推导出了点阵结构平面内等效弹性性能的解析表达式及其弹性各向同性条件。研究结果表明,正交各向异性弹性模型适用于描述熔丝制造PLA材料的弹性行为,基于该模型能够准确预测PLA材料在任意方向上的弹性模量。在预测与设计熔丝制造点阵结构的力学性能时需要充分考虑材料的各向异性。在考虑材料的各向异性之后,基于代表体元法调整点阵结构的几何尺寸,能够实现部分点阵结构的弹性各向同性设计。)
... Prior research has identified a myriad of architectures with tailored anisotropy-behaving stiff in some directions and compliant in others. Most popular for its simple fabrication and property extraction has been the class of truss lattices (13,(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28). In principle, arbitrary thermodynamically admissible stiffness combinations can be achieved by the complex arrangement of trusses (20), which may be guided by topology optimization schemes (also referred to as inverse homogenization) (13). ...
... In principle, arbitrary thermodynamically admissible stiffness combinations can be achieved by the complex arrangement of trusses (20), which may be guided by topology optimization schemes (also referred to as inverse homogenization) (13). Most prior work, however, has relied on simple parametric studies (21,29) (with a limited range of achievable [an]isotropy), structural optimization techniques (13,27) (which are expensive in three dimensions [3D] and may not guarantee manufacturability), or the selection from a precomputed UC database (14,15) (which becomes prohibitively large for high-dimensional parameter spaces). Moreover, identified topologies for different elastic stiffnesses are often incompatible and hence not continuously convertible into each other, which prevents their use in structures with spatially varying, locally optimized stiffness, such as in bone. ...
... We emphasize that our design and stiffness spaces of truss lattices (categorized into 262 unique topologies) are vastly larger than those of the closest comparable approaches (19,21,49), which examined a small set of topologies and limited the property space by varying only the relative density of elementary building blocks. As illustrated by the examples in Fig. 1 B-D, our design space includes a wide range of anisotropic (nonorthotropic) responses by combining topological and geometrical manipulations of the UC. ...
Article
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Significance More than a decade of research has been devoted to leveraging the rich mechanical playground of periodically assembled truss metamaterials. The enormous design space of manufacturable unit cells, however, has made the inverse design a challenge: How does one efficiently identify a complex truss that has given target properties? We answer this question by a data-driven method, which instantly (once trained, within milliseconds) generates not one but a variety of truss unit cells, whose effective response closely matches a given (fully anisotropic) stiffness tensor. Moreover, our framework to smoothly transition between different unit cells enables the design of lightweight structures with spatially varying, locally optimized properties, for applications from wave guiding to artificial bone.
... With the help of various analytical models of the lattice materials, the isotropic design of elastic stiffness of lattice materials has been carried out. Xu et al. [22] proposed a concept of controllable anisotropy to combine the two lattice configurations together according to a certain volume fraction so that the relative stiffness along different directions can be controlled. Isotropy in the stiffness can be achieved at a specific combination and volume fraction of lattice configurations. ...
... Design of lattice materials with isotropic stiffness ) and (21) into Eq. (20), The stress tensor in the global coordinate system is expressed as follows:σ ⎡ ⎣ cos 2 θ cos 2 ϕ cos θ cos ϕ sin ϕ cos θ cos 2 ϕ sin θ cos θ cos ϕ sin ϕ sin 2 ϕ cos ϕ sin θ sin ϕ cos θ cos 2 ϕ sin θ cos ϕ sin θ sin ϕ cos 2 ϕ sin 2 θ ⎤ ⎦ (22) Rewrite the stress tensor as the vector form, and further substitute it into the constitutive relationship Eq. (17), one can obtain the strain vector in the global coordinate system. Rewrite the strain vector as the tensor form and transform it back into the local coordinate system following Eq. ...
Article
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Lattice materials possess excellent mechanical properties such as light weight, high specific stiffness and high energy absorption capacity. However, the commonly used lattice materials inspired by Bravais lattice often give rise to property anisotropy that is not desirable for engineering application such as bone implants. For this sake, a design methodology for isotropic stiffness is proposed in this paper. Firstly, an efficient theoretical method for calculating the elastic matrices of lattice materials was presented. The method is based on Euler–Bernoulli beam theory and the assumption of affine deformation of cell vertices applicable to cubic truss-lattice materials. The theoretical approach was validated by comparing with the finite element simulations. Utilizing the validated theoretical method, and by properly combining the lattice configurations with complementary stiffness along different directions, an elastic isotropic lattice material can be obtained. A few examples are presented to demonstrate the effectiveness and adaptability of the proposed design strategy by permutating the combinations of different classic lattices. The method proposed in this paper can provide a new approach in the design of lattice materials with excellent anisotropy control.
... To design the target structure with controllable anisotropy, some strategies have been proposed and validated. The first strategy is the assembly design of different unit cell types at the same or different scales [21][22][23], and even the isotropic trusslattices could be harvested [22]. Another strategy is the replacement of truss-lattices with shell-lattices by using the bending-energy based function, and more detailed discussions could be found in [16,24,25]. ...
... However, it should be noted that, despite the anisotropy could be controlled, they all larger than 1, indicating that an isotopic solid could not be obtained in this way. An alternative way is to integrate different types of architectures into one configuration [22,23], which would be investigated in our further study. ...
Article
Truss-lattices have wide application prospects in various important fields owing to their superior mechanical properties and energy absorption characteristics. In this paper, two typical truss-lattice materials (i.e. bending-dominated body-centred cubic (BCC) and stretch-dominated Octet architectures) were designed with enhanced mechanical properties and tunable anisotropy. The elastoplastic properties and large strain response were both investigated numerically and experimentally. Numerical results showed that a relative larger stiffness characteristic could be harvested when the shape parameter was chosen between 0.2 and 0.3, and the anisotropy degree could also be controlled through the shape parameter. Large strain multi-cell simulations also demonstrated the enhanced plastic properties and energy absorption characteristics of the designed architectures. The numerical findings were then confirmed through the uniaxial compression experiments on the 316L stainless steel truss-lattices specimens fabricated by the Selective Laser Melting (SLM) process. This study would broaden the design idea for the 3D truss-lattices with enhanced mechanical performance and tunable anisotropy, which may of great potential in engineering applications.
... Another outstanding advantage of lattice structures over conventional homogeneous solid materials is the ability to achieve customizable mechanical properties from a single material through the design of structural morphology at the microscale [10]. For example, the precise control of anisotropy of a lattice material can be achieved by designing its structural form [11]. ...
... Compared with the studies on the stiffness, strength, and energy absorption properties of lattice structures, researchers have paid less attention to their anisotropy control. Recently, Xu et al. [11] proposed a controllable anisotropic lattice structure design strategy, of which the core concept is to control anisotropy by changing the rod-diameter ratio of the unit structure, which can obtain an isotropic lattice structure. Similarly, Tancogne et al. [12] and Latture et al. [13] developed lattice structures with isotropic elastic properties by combining typical lattices. ...
Article
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Controllable anisotropy of the microscale lattice metastructures is the key to achieving designable mechanical properties of the macrostructures. However, a current computational challenge is precisely controlling the anisotropy of the design domain containing an ultra-large number of lattice structural units. In this work, we decompose the large-scale design domain into several mesoscale subregions and propose an artificial intelligence-aided design method that can automatically control the anisotropic properties of the mesoscale design domain. First, a 3D convolutional neural network is proposed to replace the time-consuming homogenization computation process, thus enabling fast prediction of equivalent mechanical properties of lattice structures with arbitrary geometrical and material parameters. Then, inspired by the encoded metaplot, the mesoscale design domain is divided into several subunits with certain encoding. Finally, the combination type of the lattice structures of the subunits is adjusted automatically by a genetic algorithm according to the given target anisotropic properties. Furthermore, the effects of the lattice types, subunit numbers, and species number of subunits contained in the subregion on the accuracy of the inverse design are discussed. The results show that the proposed intelligent design method can effectively control the anisotropic properties of the mesoscale design domain.
... In both cases, the bottom face of the structure was attached to the substrate and fully constrained. This methodology has been employed before for the precise calculation of the polar stiffness map of structure, leading to a close correlation between theoretical and experimental results [69]. Further information on the applied boundary conditions and FEA models can be found in SI. ...
... The various maps are presented in Figure 9. To quantify the variance of the stiffness for the different structures, the Zener ratio A is used [69], defined as ...
Article
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Using a new Bayesian Optimization algorithm to guide the design of mechanical metamaterials, we design nonhomogeneous 3D structures possessing the Cauchy symmetry, which dictates the relationship between continuum and atomic deformations. Recent efforts to merge optimization techniques with the design of mechanical metamaterials has resulted in a concentrated effort to tailor their elastic and post elastic properties. Even though these properties of either individual unit cells or homogenized continua can be simulated using multi-physics solvers and well established optimization schemes, they are often computationally expensive and require many design iterations, rendering the validation stage a significant obstacle in the design of new metamaterial designs. This study aims to provide a framework on how to utilize miniscule computational cost to control the elastic properties of metamaterials such that specific symmetries can be accomplished. Using the Cauchy symmetry as a design objective, we engineer structures through the strategic arrangement of 5 different unit cells in a 5 × 5 × 5 cubic symmetric microlattice structure. This lattice design, despite constituting a design space with 5^10 3D lattice configurations, can converge to an effective solution in only 69 function calls as a result of the efficiency of the new Bayesian optimization scheme. To validate the mechanical behavior of the design, the lattice structures were fabricated using multiphoton lithography and mechanically tested, revealing a close correlation between experiments and simulated results in the elastic regime. Ultimately, a similar methodology can be utilized to design metamaterials with other material properties, aspiring to control properties at different length scales, an endeavor that requires inordinate computation cost.
... Lattice structure belongs to cellular structures that are widely adopted by nature in biological organisms such as bones and wood [127]. The main motivations of using lattice structures for ASD design are: 1) the ability of lattices to conform to a freeform 3D space [128] and 2) the ability of lattices to exhibit a controlled anisotropy through topology variation [129]. Additional motivation is the conclusion drawn in Chapter 4 that the structure of an ASD is crucial for natural anisotropy restoration. ...
... Despite the potential of lattice structures, there is no systematic and effective methodology for the design of such lattice structures to achieve a controlled anisotropy [129]. The conventional method is to select one or several unit-cells and make modifications on them, which is time-consuming as the whole process is based on trial-and-error while the existence of the desired anisotropy is not guaranteed [134]. ...
Thesis
Low back pain, which is a worldwide health problem, is the leading cause of activity limitation and work absence throughout much of the world and causes an enormous economic burden on society [1][2]. Literature shows that 70–85% of all people will have back pain at some time in life [3]. Compared to spinal fusion, total disc replacement (TDR) that uses an artificial spinal disc (ASD) to replace the degenerated intervertebral disc (IVD) is a motion-preserving surgical technique to treat serious back pain. Although ASD design is shown to be critical to the clinical outcomes of TDR and many attempts have been made to improve ASD designs, very little success has been achieved. The development of a reliable ASD that is able to restore the natural mechanics of the implanted spinal segment remains to be a substantial challenge, which is mainly due to the complicated biomechanical environment faced by the IVD and the complex mechanics of an IVD that consists of nonlinear, viscoelastic, and anisotropic behavior [4]. Inspired by the potential of second-generation viscoelastic ASDs and the design freedom provided by multi-material additive manufacturing (AM), this thesis explores the feasibility of using an elastomeric, multi-material, patient-specific ASD fabricated with multi-material AM for achieving anatomical match and reproducing customized mechanics. Five conceptual designs for multi-material ASD that either employ biomimicry or structures inspired by mechanical metamaterials are proposed. This thesis also proposes a general computational design method for the optimization of patient-specific ASDs to reproduce customized mechanics. The performance of five proposed conceptual designs is investigated using in-vitro mechanical testing to determine the most favorable design concept according to defined design criteria. Given this design concept and the manufacturing constraints, the overall workflow for patient-specific ASD design is implemented to examine the ability of this design concept to reproduce patient-specific mechanics. To improve the efficiency of the computational design method, the feasibility of several strategies such as problem dimension reduction and a regression model is also explored in this work. In the end, general guidelines for the design of multi-material ASDs are proposed from the knowledge gained throughout this thesis. This thesis shows that ASD design concepts mimicking the structure found in an IVD have great potential for restoring natural mechanics. In addition, ASD design concepts that employ structures inspired by mechanical metamaterials are able to exhibit nature-mimicking nonlinear behavior that are otherwise difficult to achieve with the given materials and 3D printing technique. Furthermore, this thesis verifies the general applicability of computational method for patient-specific ASD design and strategies such as machine learning for improving the efficiency of the design process. This thesis demonstrates the great potential of a patient-specific, multi-material ASD fabricated with multi-material AM, and serves as inspiration for the future development of multi-material ASDs.
... Lattice structures have significant advantages, such as being structurally lightweight, showing good thermal conductivity, and enhancing impact protection [1][2][3][4]. It has been widely used in many applications, for instance, in aerospace [5], machinery [6], medicine [7,8], and other fields [9,10]. However, lattice structures with controllable volume fractions and well-defined internal architectures are not easy to fabricate by traditional fabrication processes, which is the biggest problem encountered in their development [11]. ...
... In the literature, Lee et al. [51] found that the elongated G structure can significantly increase compressive strengths. In this study, Eq (5) was rewritten so that the geometrically lengthened or shortened P and G structures can be designed by changing the value of m z in Eq (8): ...
Article
Triply periodic minimal surface (TPMS) structures have been extensively investigated for their excellent mechanical properties and lightweight potential. In this study, a new hybrid gradient (HG) TPMS structure was proposed by combining geometrically deformed gradient (GDG) and volume fraction gradient (VFG). All designed structures were fabricated by selective laser melting (SLM) using Ti-6Al-4V. They were investigated experimentally in terms of deformation behavior, stress–strain curve and energy absorption. The results demonstrated that the GDG structure can develop a better deformation mode to enhance the energy absorption, while the VFG structure can help to reduce the initial peak force and delay the densification point. Most importantly, the hybridization of these two gradients can both greatly enhance the energy absorption capacity and delay the densification point. Compared with the uniform structures, the hybrid gradient structures can almost double the energy absorption under compression. The results of this research may provide valuable insights for the design of high-performance energy-absorbing structures in the future.
... Applications of lattice materials in research fields like scaffolds for bone regeneration [19], substrates for tissue growth [52], multi-lattice structures [53], and metamaterials [1] have set additional requirements to designing a materials' anisotropic properties [54]. Simulations approaches have been used to plot the anisotropic modulus map of periodic lattices and composites [55,56]. ...
... Simulations approaches have been used to plot the anisotropic modulus map of periodic lattices and composites [55,56]. It has been shown that truss-based lattice materials often exhibit high mechanical anisotropy; introducing redundant trusses and combining different lattices types provide one efficient way to adjust anisotropy and to achieve quasi-isotropic 4 properties [54]. Designing the structural morphology provides an alternative approach to tune anisotropy [57]. ...
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Obtaining multifunctionality from microstructure instead of constituents provides a new direction for developing multifunctional materials. Periodic hollow sphere foam (HSF) offers one lightweight structural motif with integrated open and closed cells, high energy absorption, low thermal conductivity, and triple-negative material indices. Here, we investigate the direction-dependent mechanical property, instability, and elastic wave isolation behavior of HSFs. Explicit formulas, stereographic projections, and general scaling relationships are developed to quantify and visualize the anisotropic mechanical properties of HSFs. By investigating the directional wave propagation in HSFs, extremely wide phononic band gaps are discovered in the HSFs. The derived formulas and the simulation-informed parametric maps allow us to design HSFs with desired static and dynamic anisotropic property profiles, including tailorable direction-dependent stiffness/shear modulus, negative Poisson's ratio, and wave isolation properties. Building upon these results, multifunctional design concepts of HSFs are further set forth. This study not only reveals tailorable mechanical anisotropy and band gap in HSFs, but also developed a general approach to investigate the direction-dependent properties of periodic materials, enabling multifunctional applications where lightweight, direction-dependent property, wave attenuation, and programmability are required simultaneously.
... Truss lattices are a group of open-cell lattice structures; through appropriate design of the connectivity of elementary trusses, they can exhibit stretching-dominated deformation behaviors and higher specific stiffness and strength than bending-dominated counterparts [12][13][14]. However, the truss members can only bear uniaxial loadings, thus leading to lower stiffness and strength than the theoretical upper bounds [15][16][17][18]. In contrast, open-cell shell lattices, which are composed of smooth and non-intersecting thin shells, possess superior stiffness and strength than truss lattices with equal relative densities since shells bear external loads by in-plane stresses [19,20]. ...
... where and represent the two Lagrange multipliers. The material derivative is derived by Eq. (13), (16) and (18) as: ...
Article
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Shell lattices are composed of smooth, non-intersecting and periodic thin shells. Their open-cell topology facilitates the manufacturing and multifunctional applications. This work proposes a shape optimization framework to obtain uniform thickness shell lattices with superior isotropic elasticity. A B-spline parameterized Monge patch model is used to represent the mid-surface within the 1/48 unit cell, which maintains the cubic symmetry and simplifies the sensitivity evaluation. Two groups of elastically isotropic shell lattices are obtained, including Primitive (P) and I-graph-wrapped package (IWP). The highest achievable bulk, Young’s, shear moduli of P/IWP family lattices are nearly 70%/80%, 40%/60%, 40%/60% of the Hashin-Shtrikman upper bounds at 10% relative density. Besides, the Young’s/bulk modulus maximization is further introduced into the optimization to seek potential improvement of stiffness, which yields similar optimized lattices with close stiffness for arbitrary initial designs. The highest achievable elastic moduli are slightly improved by 3∼5% than those without moduli maximization. In general, P-family lattices possess comparable Young’s, shear and higher bulk moduli than the stiffest truss lattices, while IWP-family lattices possess superior stiffness. This work proposes a systematic design approach to obtain elastically isotropic uniform thickness shell lattices, which can be applied to the other lattice families with Monge patch representations.
... This implies that the isotropic lattice structure can be obtained by controlling the w for all volume fractions. Therefore, in addition to the traditional methods for controlling the anisotropy of lattice structures such as adjusting the strut diameter [55,56], designing the hollow struts [57,58], and adjusting the surface morphology [47,59], designing the interpenetrating lattice structures provides a new way to control the structural anisotropy. ...
... Xu et al. [30] studied a method for designing a cellular structure using anisotropic values as a user input. This cellular structure is basically lattice-based. ...
Article
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Three-dimensional (3D) printers enable the realization of parts with complex shapes, particularly parts with an internal structure as well as an external shape. They are beginning to be used in real production as well as for prototyping. Because 3D printers allow shapes for parts that could not be imagined when conventional machining processes were the only fabrication methods, designers need to design porous parts with minimum weights that can maintain the required load-bearing capability without concerns about the realization of those parts. However, designers cannot design the internal structure of the parts using the current computer-aided design systems as freely as they can design the external shape. Therefore, in this paper, an intuitive design tool is proposed for users to interactively design an internal structure inside given external shape. To create a porous structure inside given closed volume, we aimed to generate a honeycomb-like structure comprising cells whose size and crowdedness can be intuitively manipulated by a designer. Thus, fine cells exist in certain regions, while coarse cells exist in the remaining regions according to the design. To realize this aim, the phenomenon in which more iron particles are attracted near a magnet and fewer are attracted further away from the magnet is imitated. More and finer honeycomb-like cells are attracted near a magnet located on the external surface of the closed volume. The designer can add magnets and move them on the external surface until the desired internal honeycomb-like structure is obtained.
... In addition, the Hashin-Shtrikman (HS) upper bounds, which exhibited the largest theoretically achievable elastic characteristics of an isotropic porous solid were also taken into account. The HS upper bounds comprised the bulk modulus (K HSU ), shear modulus (G HSU ) and Young's modulus (E HSU ) [58]. ...
Article
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Additive manufacturing (AM) significantly provides opportunities to design and produce components with complex shapes and geometries without a demand for many conventional manufacturing steps. The AM technique has been also used for accomplishing such customized, multi-cell lattice structures for lightweight applications. However, an eligible design approach for integrating lattice structures is challenging so that desired mechanical properties in various advanced engineering structures can be achieved. Therefore, multi-cell lattice structures and graded design approaches for improving load-bearing capacities of additively manufactured parts were studied in this work. C-clip structures infilled with different lattice configurations were investigated under tension load. It was found that uniform BCC-Shell and SC-BCC-Shell lattice samples exhibited considerably higher overall stiffness values than the others due to their large normalized shear moduli. By both stress- and topology optimization-based tailored density methods, lattice structures with suitable effective properties, anisotropic characteristics, and relative densities could be employed, especially for highly loaded regions. The proposed strategies provided the structural performance which was just less than 3% from the benchmark.
... LSs exhibit superior and customizable mechanical properties, making them appropriate for application in aeronautics, automotive, marine structural, transportation, defense, and biomedical components, e.g., implants, artificial bones, and scaffolds for tissue engineering [14][15][16][17][18][19]. These properties include ultra-lightweight, high specific stiffness and strength, excellent energy absorption capability, high fracture toughness, and damping [15,[20][21][22][23][24]. ...
Article
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With the increasing development of additive manufacturing (AM) technology, lattice structure (LS) emerged and expanded as a subset of cellular materials. LSs' mechanical properties mainly depend on the relative density, the unit cell topology, the manufacturing processes, and the base material. In this research, PA12 lattice structures with non-auxetic strut-based topologies, including BCC, FCC, FCCz, FBCC, FBCCz, FBCCxyz, and OT, were manufactured by selective laser sintering (SLS) and were tested under quasi-static compression. Data from the compression test was analyzed and investigated to achieve mechanical properties such as strength, elastic modulus, and absorbed energy. OT has the highest yield strength (4.07 MPa), ultimate strength (4.53 MPa), specific ultimate strength (10.11 MPa), elastic modulus (0.099 GPa), specific elastic modulus (0.221 GPa), and plateau stress (9.98 MPa) among the investigated sturt-based topologies. BCC has the lowest properties. The absorbed energy (W) for OT and FBCCxyz is higher than in other topologies. FBCCz has the highest volumetric energy absorption (WV) (0.284 MJ/m 3) up to the strain of the UTS point, and FCCz has the lowest (0.152 MJ/m 3). The finite element method (FEM)-based ABAQUS software was used to simulate the behavior of LSs under compression test. Also, SEM micrographs of struts fractured surfaces in the CP lattice block were investigated. The failure mechanism in most LSs is the layer-by-layer failure of rows in the strut-based structure. According to finite element modeling results, stress concentration occurred in the nodes and adjacent areas, making cracks, and fractography exhibited ductile fracture in these regions.
... By applying ceramic stereolithography, complex cellular ceramics with lattice structures can be produced [23][24][25][26] . In previous studies, different lattice configurations were designed and tested to obtain the optimized mechanical behaviors [27][28][29] . Furthermore, since heat is released during the polymerization of ceramic suspensions with volume shrinkage, it was also found that the accumulation of shrinkage exists as one of the main factors causing the deformation and cracking of the stereolithography-fabricated ceramic parts [30,31] . ...
Article
Ceramic cores are widely used in investment casting, and ideal properties of cores are essential for high-quality castings. Under the circumstances requiring thick cores, solid cores are likely to encounter deformation and cracking defects due to the accumulation of shrinkage. Therefore, with the superiority of ceramic stereolithography in producing complex ceramic parts, hollow cores with lattice structures were designed and fabricated. The dimensional accuracy and properties of the green and sintered bodies were evaluated. Results show the dimensional accuracy of sintered cores is controlled within ±0.25 mm benefited from the precise green bodies. The mechanical properties are not obviously deteriorated. The bending strength reaches 11.94 MPa at room temperature and 12.87 MPa at 1,500 °C with a creep deformation of 0.345 mm. Furthermore, casting verifications prove that the hollow cores meet the requirements of investment casting. Smooth casting surfaces are obtained, at the same time, the core-removal efficiency is improved by over 3 times.
... A simple procedure of numerical homogenization by Finite Elements is applied. This method proposed by Steven [28] has been adapted to lattices [29]. For that, the stiffness matrix C ijkl is obtained in the elastic domain. ...
Article
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To obtain a functional part from additive manufacturing (AM) technologies, some surfaces require post-processing by machining. An approach is developed using additive manufacturing supports as a clamping device for the milling operation. A model combining an analytical approach to determine the cutting forces with a finite element model (FEM) to predict the dynamical response of the workpiece-supports system is proposed. The complex structure of the supports is homogenized with a simplified geometry with equivalent stiffness and mechanical properties. A case study from the biomedical field is proposed: the finishing operation of a custom-made maxillary reconstruction plate is simulated. A parametric study is proposed with: (1) two different lattice geometries used as support structures; (2) up and down milling; (3) different depths of cut.
... Studies on the design and production of rigid, strong, and light truss structures are increasingly continuing. The design of unit structures in repetitive order is critical in some applications [24]. ...
Article
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As a result of millions of years of evolution, nature has created structures that are resistant to the environment it is in, and these structures have inspired people to solve problems. However, these structures found in nature are very difficult to manufacture due to their complex architecture. At this point, with the increasing interest and development in additive manufacturing (AM) technologies, these inspired structures have become applicable. In order to provide design freedom that traditional manufacturing methods cannot provide, the design for additive manufacturing (DFAM) design model has emerged, depending on the capacity of additive manufacturing technologies. In this design model, designs with high mechanical properties can be created by using lattice structures and topological optimization methods. The combination of these two methods leads to the creation of bio-inspired designs. Thus, bio-inspired, lightweight and high-strength designs can be developed. In this study, a combination of biomimetic approaches and additive manufacturing design was taken. This is an open access article under the CC BY-SA 4.0 license. (https://creativecommons.org/licenses/by-sa/4.0/)
... Moreover, it has been shown that certain polygonal-based two-dimensional unit-cell designs can yield Poisson's ratio values which are well above the common engineering isotropic material limits and within the range encountered for ligaments and tendons (Karathanasopoulos et al., 2017b). The metamaterial's effective elastic stiffness properties can be tuned through modifications of the unit-cell inner structural design, so that the effective mechanical attributes obtained in each material direction differ by orders of magnitude (Karathanasopoulos et al., 2018)- (Xu et al., 2016). ...
Article
The engineering of tendon and ligament tissue biocompatible restoration materials constitutes a long-standing engineering challenge, from the chemical, biological and mechanical compatibility analysis and design perspective. Their mechanics are inherently anisotropic, exceeding the potential limits of common, non-architected engineering materials. In the current contribution, the design of advanced material or “metamaterial” architectures that can emulate the mechanical properties observed in native tendon and ligament tissues is analytically, experimentally, and numerically investigated. To that scope, anisotropic metamaterial designs that are based on rectangular cuboid architectures with and without inner body-centered strengthening cores are considered. Thereupon, the metamaterial design specifications required for the approximation of the highly anisotropic tissue performance, namely of the characteristic elastic, shear, and Poisson's ratio attributes are studied. It is shown that certain strengthened, anisotropic body-centered cuboid lattice architectures allow for substantial effective metamaterial stiffness along the primal tissue loading direction, upon a rather low shear loading resistance. The previous mechanical attributes come along with Poisson's ratio values well above unity and moderate relative density values, furnishing a combination of material characteristics that is highly desirable in restoration praxis. The analytically and numerically guided anisotropic metamaterial performance is experimentally reproduced both for the case of uniaxial and shear loads, using a microfabrication stereolithography additive manufacturing technique. The obtained scanning electron microscopy images highlight the fabrication feasibility of the identified metamaterial architectures, in scales that are directly comparable with the ones reported for the natural tissues, having feature sizes in the range of some 10ths of micrometers and elastic attributes within the range of clinical observation.
... This structure is comparable to the lattice structure that is seen in crystals. Among the various material architectures, lattice structures are capable of achieving the highest structure efficiency per unit weight [71]. As a result of this, lattice structures are widely used in applications that place a premium on minimizing weight, such as aerospace engineering and automobile manufacturing [72]. ...
Article
Lattice structures are vital for biological applications because of its numerous benefits (for example, faster and stronger binding to bone tissue). Consequently, processing of lattice structure is a particularly popular area of study currently. In this study, additive manufacturing technologies utilized in several engineering disciplines were collated and their merits and shortcomings were examined. Numerous sectors and disciplines view lattice structured additive manufacturing as a prototyping technique. In recent years, additive manufacturing technology has also progressed toward the fabrication of useable final goods. The objective of this review is to classify the produced systems under the headings of aviation, automotive, and military technologies within the context of engineering and to compare them by examining the research and technology firms in this sector. In this categorization, lattice-structured additive manufacturing techniques are categorized as an engineering production technology, and examples of this field are investigated. Technologies, which are examples of diverse engineering applications, are categorized under four primary headings: additive manufacturing knowledge, selective laser melting (SLM), lattice structure, and changeable porosity cellular structures.
... Lattice structures are usually elastically anisotropic, with directions which are weaker in relation to the arrangement of edges. This anisotropy is highly dependent on the exact spatial arrangement of the edges so that the elasticity must be evaluated for numerically [20]. ...
Preprint
In certain quantum gravity theories known as emergent gravity theories, it is proposed that spacetime is divided up into a discrete structure at the Planck scale. Such theories are likely ruled out by the failure to observe violations of Lorentz invariance at such length scales. Another reason to doubt these theories is the fact that one would expect the appearance of lattice defects in the discrete lattice (especially edge dislocations and stacking faults). It should be possible in principle to observe these dislocations, although one must use gravitational waves of a frequency which cannot naturally be produced in the Universe. We also raise the possibility that these discrete lattices should have physical anisotropies, where the most obvious choice for such anisotropies are also ruled out by experimental bounds.
... It draws our attention that the particular topology of a lattice impacts in a high degree its apparent mechanical properties. This can be viewed as a great opportunity to tune locally these mechanical properties in order to mimic the mechanical response of a bone to be replaced [9], [35]. Moreover, the relative density of the lattices, or the so-called volume fraction, plays an important role in the control of their apparent mechanical properties [36]. ...
Thesis
Les structures lattices offrent des solutions performantes pour remplacer des structures tout en contrôlant leurs propriétés mécaniques. D'un point de vue mécanique, ceci est particulièrement le cas pour réduire le phénomène de stress-shielding qui peut survenir dans l’os péri-implantaire suite à la pose d’implants. Outre les structures lattices à base de poutre largement étudiées, le concept de surfaces minimales triplement périodiques (TPMS) a été introduit pour créer des structures présentant des propriétés originales. Huit topologies de cellules unitaires basées sur TPMS et deux basées sur des poutres ont été étudiées numériquement pour calculer leurs propriétés élastiques effectives et la distribution des contraintes locales, en utilisant une méthode d'homogénéisation périodique. La dépendance de la topologie sur la distribution des contraintes locales est mise en évidence par une analyse statistique des cellules unitaires. La validation des résultats numériques a été effectuée par des résultats obtenus expérimentalement suite à la fabrication par une machine SLM de plusieurs structures et à des tests de compression. Deux types de sollicitation ont été étudiés : un cas fonctionnel mettant l'accent sur le module d'Young effectif des structures, et un cas destructif se concentrant sur la rupture de ces dernières. Sur la base de ces résultats, une étude numérique considérant le remplacement partiel d'un os par une structure lattice à base de TPMS, soumis à des conditions aux limites réelles, est développée. Un critère permettant de choisir une structure optimisée sur la base de contraintes mécaniques est proposé.
... Note that, while there are elastically isotropic crystalline architectures in 2D (e.g. the triangular or honeycomb lattice studied here), this is no longer the case in 3D [42]. Lattice structures with isotropic elasticity are important for many applications and, as such, are the subject of several works [26,[46][47][48][49]. Introducing disorder in a tunable way as proposed here may offer a promising route to this aspect. ...
Article
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We examine how disordering joint position influences the linear elastic behavior of lattice materials via numerical simulations in two-dimensional beam networks. Three distinct initial crystalline geometries are selected as representative of mechanically isotropic materials with low connectivity, mechanically isotropic materials with high connectivity, and mechanically anisotropic materials with intermediate connectivity. Introducing disorder generates spatial fluctuations in the elasticity tensor at the local (joint) scale. Proper coarse-graining reveals a well-defined continuum-level scale elasticity tensor. Increasing disorder aids in making initially anisotropic materials more isotropic. The disorder impact on the material stiffness depends on the lattice connectivity: Increasing the disorder softens lattices with high connectivity and stiffens those with low connectivity, without modifying the scaling between elastic modulus and density (linear scaling for high connectivity and cubic scaling for low connectivity). Introducing disorder in lattices with intermediate fixed connectivity reveals both scaling: the linear scaling occurs for low density, the cubic one at high density, and the crossover density increases with disorder. Contrary to classical formulations, this work demonstrates that connectivity is not the sole parameter governing elastic modulus scaling. It offers a promising route to access novel mechanical properties in lattice materials via disordering the architectures.
... Note that, while there are elastically isotropic crystalline architectures in 2D (e.g. the triangular or honeycomb lattice studied here), this is no longer the case in 3D [42]. Lattice structures with isotropic elasticity are important for many applications and, as such, are the subject of several works [26,[46][47][48][49]. Introducing disorder in a tunable way as proposed here may offer a promising route to this aspect. ...
Preprint
Full-text available
We examine how disordering joint position influences the linear elastic behavior of lattice materials via numerical simulations in two-dimensional beam networks. Three distinct initial crystalline geometries are selected as representative of mechanically isotropic materials low connectivity, mechanically isotropic materials with high connectivity, and mechanically anisotropic materials with intermediate connectivity. Introducing disorder generates spatial fluctuations in the elasticity tensor at the local (joint) scale. Proper coarse-graining reveals a well-defined continuum-level scale elasticity tensor. Increasing disorder aids in making initially anisotropic materials more isotropic. The disorder impact on the material stiffness depends on the lattice connectivity: Increasing the disorder softens lattices with high connectivity and stiffens those with low connectivity, without modifying the scaling between elastic modulus and density (linear scaling for high connectivity and cubic scaling for low connectivity). Introducing disorder in lattices with intermediate fixed connectivity reveals both scaling: the linear scaling occurs for low density, the cubic one at high density, and the crossover density increases with disorder. Contrary to classical formulations, this work demonstrates that connectivity is not the sole parameter governing elastic modulus scaling. It offers a promising route to access novel mechanical properties in lattice materials via disordering the architectures.
... The less anisotropic properties of lattice structures with taper struts were attributed to the improved mechanical properties in certain directions (such as [100] direction) and weakened mechanical properties in other directions (such as [111] direction). In previous studies, various Anisotropy indexes were obtained by combining basic lattice units [42,43] or changing the connections of struts [44]. Although these methods can obtain lattice structures with a large range of Anisotropy indexes, the topology configuration of lattice unit cells could be modified by them, which could influence the connections of structures. ...
Article
The fractures of lattice structures are mainly near the nodes during loading due to the stress concentration, resulting in low mechanical properties. In this study, a novel parametric approach was proposed for modelling Body-Centered Cubic (BCC) lattice structures with taper struts to reduce the stress concentration at nodes and improve the mechanical properties. The mechanical properties and deformation behavior of lattice structures with different taper struts were investigated through finite element analysis and uniaxial compression tests. The simulation results demonstrate that tapered struts significantly reduced the anisotropy of BCC lattice structures and increased the elastic modulus by up to 67%, while the shear modulus was only improved for a slight tapering by less than 2.92%. In addition, the fracture occurred due to the combination of high plastic strain and tensile stress, and the fracture site of BCC lattice structures changed from the node region to the center of strut due to the tapering of the strut cross-sections, which corresponded to the experimental results. Finally, an optimization strategy was developed to simultaneously optimize the distribution of volume fraction and strut geometry of BCC lattice structures, demonstrating the effectiveness of the proposed design and optimization method for lightweight applications.
... For the mathematical description, different types of mathematical fits are compared considering their suitability for the simulated curves as well as for the physical interrelations of comparable solid materials. The following types, which have already been considered in literature, are looked at and exemplary shown for the FCC strut lattice in comparison inFigure 4(Cheng et al., 2018;Cheng et al., 2017;Vega-Moreno et al., 2020;Xu et al., 2016;Nguyen et al., 2021;Crupi et al., 2017;Ruiz de Galarreta et al., 2020):Polynomial fit: ( ) = 0 + 1 + 2 2 + ⋯ + n of first to fourth order Power fit: ( ) = Exemplary comparison of different mathematical fits for FCC strut lattice cell ...
Article
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AM-meso structures offer a high potential for adapted properties combined with lightweight design. To utilize the potential a purposeful design of the meso structures is required. Therefore, this contribution presents an approach for modelling their properties depending on design parameters by scaling relationships. The relationships are investigated based on grey box and axiomatic models of elementary cells. Exemplary the pressure stiffness is determined using FEM in comparison to an analytical approximation. The comparison reveals effects and influences occurring within the elementary cell.
... The basic mechanical properties of the four cells selected in the article have been analyzed in detail in our previous research and the stiffness matrix is calculated [51]. The equivalent mechanical properties in each direction of lattice structures are calculated, which are anisotropic [36,49]. The 3D space representations of effective Young's modulus surfaces were estimated by means of coordinate transformation of the modulus and Poisson's ratio of each unit cell obtained by finite element calculations (Table 1) [50,51]. ...
Article
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Adjusting the mechanical properties of lattice structures is important for many modern application fields. In this paper, a new design method for hybrid multi-layer lattice structures was developed to improve the mechanical properties and energy absorption, by altering and suppressing the formation of shear band. In these hybrids, all unit cells were divided into two parts: i) diagonal unit cells and ii) matrix unit cells. Four categories of unit cells were selected to construct the hybrid multi-layer structures. The compressive moduli, ultimate strengths, and energy absorption properties of the laser powder bed fusion (L-PBF) fabricated structures were assessed by experiments and finite element analysis (FEA). The results revealed the great impact of diagonal unit cells on the mechanical properties of the structures. Stronger diagonal unit cells than matrix unit cells led to hybrid structures with enhanced mechanical properties. Compared to uniform BCC lattice structure, the relative density of the lattice structure consisting of the weakest BCC matrix unit cells and strongest BFVC diagonal unit cells (coupling of BCC, FCC, and VC) exhibited an increase of20%. The compressive modulus and ultimate strength of this structure rose by more than 200% and 90%, respectively. Two types of structures with specific properties were generated by hybrid design.The first displayed higher modulus, superior strength, and elevated specific energy absorption(SEA) but lower crash load efficiency (CLE). The second illustrated simultaneously higherSEA and elevated CLE. The present results provide a new insight for improving the load-bearing and energy absorption capacities of lattice structures.
... This effect has a significant impact on the stress concentration on the structures, which may cause early failure and catastrophic collapse. To quantify the anisotropy of structures, the 3D polar stiffness map needs to be calculated and plotted numerically [11,50,51]. Since most metamaterial designs possess cubic symmetry, anisotropy index A is often used as a measure to quantify the anisotropy 4,8 . ...
Article
Architected metamaterials are built upon the assembly of repeating cellular structures, exhibiting unprecedented mechanical properties attributed to the tunability of cellular geometries. They have demonstrated a wide range of applications in the optical and electromagnetic fields, and recently they are employed as advanced tissue engineering scaffolds. The microgeometry-driven strategy enlarges the design domain of scaffold features and enables more accurate manipulation of cell-material interactions. In this review, we introduce the most popular metamaterial designs in biomedical engineering and summarize their representative applications to fabricate in vitro models and in vivo implants. These studies validate the unique advantages of metamaterials in supporting mechanobiological studies and improving the functionality of tissue replacements. Nevertheless, the evolution of meta-biomaterials requires more detailed investigations of the relationship between structural designs and cell phenotypes as well as accurate theoretical models.
... Besides, the anisotropy of LUCs is one of the key factors when synthetically evaluating the overall performance of FGCS as it will influence the distribution of LUCs within FGCS. Thus, the Zener ratio and Young's modulus surface are also presented in Fig. 4 [36,44,45]. The anisotropy of LUCs can be quantified by Zener anisotropy index A which can be computed by Eq. (22). ...
Article
Cellular structures in nature have attracted great attention in the field of structural optimization. This paper proposes a Kriging-assisted topology optimization method for design of functionally graded cellular structures (FGCS), which are infilled by smoothly-varying lattice unit cells (LUCs). Specifically, LUCs are depicted by level set functions and the shape interpolation method is employed to generate sample LUCs. Then, one Kriging metamodel is constructed to predict the mechanical properties of LUCs within FGCS, so as to reduce the computational expense involved in finite element analysis of LUCs. Meanwhile, the other Kriging metamodel is created to predict the values of shape interpolation function, and a Kriging-assisted morphological post-process method is put forward to achieve the smooth transition between adjacent graded LUCs. In the proposed method, the effective densities, mechanical properties, and geometrical configurations of LUCs are coupled by Kriging metamodels, so that the multiscale design of FGCS can be realized at a low computational burden by optimizing the distribution of elemental densities, and this also paves the way for design of FGCS with irregular geometries by morphological post-process and geometry reconstruction. Numerical examples are presented to validate the accuracy and effectiveness of the proposed method for design of FGCS. What is more, the design of a pillow bracket with a slightly complex geometry is provided to illustrate the engineering application of the proposed method. The results indicate that the proposed method is effective and universal for designing FGCS with smoothly-varying LUCs.
Article
Pattern transformation in a periodic porous structure has inspired multifarious mechanical metamaterials/metastructures due to the induced unusual negative Poisson's ratio behavior of macroscopic materials. Recently, it has been leveraged to architect a variety of designable and multifunctional structural members. Inspired by this design methodology, a novel porous cylindrical shell, which is perforated by a large number of staggered openings, is constructed and investigated meticulously. A stable, anti‐disturbed, and controllable waisted deformation of the architected cylindrical shell will be triggered under an axial compression. A stoma‐shaped biomimetic hole and graded distribution of initial openings are proposed to ensure that the holes distributed throughout the shell can be closed up concurrently while the closed states of holes can be flexibly programmed. To explore the applications of such shells, a handy cylindrical vessel is elaborately designed and its multiple functions including reagent release, underwater sampling, and flow control are exhibited by experiments. The results reflected that the designed vessel can be facilitated with many advantages such as uniform release, quick action, easy actuation, and repeated usage. Moreover, it also might open a new avenue for metamaterials in the fields of biomedical engineering, underwater detection, fluid machinery, etc. A novel porous cylindrical shell is constructed based on pattern transformation. A stoma‐shaped biomimetic hole and graded distribution of openings are proposed to program the closing states of holes. A cylindrical vessel with arbitrary closing state is established by programming the distribution of holes. Multi‐functional cylindrical vessel can be applied for reagent release, underwater sampling, and flow control.
Article
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Architected materials are increasingly applied in form of lattice structures to biomedical implant design for the purpose of optimizing the implant’s biomechanical properties. Since the porous design of the lattice structures affects the resulting properties of the implant, its parameters are being investigated by numerous research articles. The design-related parameters of the unit cells for a strut-architected material are mainly the pore size and the strut thickness. Until today, researchers have not been able to decide on the perfect values of the unit cell parameters for the osseointegration process and tissue regeneration. Based on in vivo and in vitro experiments conducted in the field, researchers have suggested a range of values for the parameters of the lattice structures where osseointegration is in acceptable status. The present study presents a comprehensive review of the research carried out until today, experimenting and proposing the optimum unit cell parameters to generate the most suitable lattice structure for the osseointegration procedure presented in orthopedic applications. Additional recommendations, research gaps, and instructions to improve the selection process of the unit cell parameters are also discussed.
Article
To tackle the challenge in artificial spinal disc (ASD) design of restoring the mechanics of a natural disc, this study proposes an innovative lattice-based ASD for reproducing a patient-specific anisotropic rotational response, inspired by the design freedom provided by lattice structures. Motivated by the great potential of machine learning to improve computational design processes, a method is proposed for computationally efficient topology optimization using artificial neural networks (ANNs) and a subsequent member sizing for automating the design of patient-specific ASDs. The results reported in this study show a good match between the optimized ASDs’ six rotational stiffnesses with those of both L2-L3 and L4-L5 human lumbar discs. Additionally, the fast convergence rate of the optimization verifies the application of ANNs and the proposed strategy to reduce the design space by formulating the design problem as optimizing the unit-cell distribution in a predefined grid. Therefore, the study demonstrates that a lattice-based ASD is able to reproduce patient-specific anisotropic rotational response and that a machine-learning-based method improves the computational efficiency of an automated design process to produce personalized ASD designs.
Article
Lattice-like materials featuring periodic planar tessellation of regular rigid blocks connected by linear elastic interfaces and chiral or achiral properties are considered. The chirality results from a uniform rotation of the blocks with respect to their centroidal joining line and leads to interesting auxetic and dispersive acoustic behaviors. The governing equations of the discrete Lagrangian model are properly manipulated via the novel enhanced continualization scheme in such a way to obtain equivalent non-local integral and gradient-type higher-order continua. Based on the formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions accounting for shift operators and proper pseudo-differential downscaling laws, the proposed enhanced continualization technique allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall non-local constitutive tensors, as well as qualitative agreement and quantitative convergent matching of the actual complex frequency band structure. The theoretical findings are successfully verified though the study of wave dispersion and attenuation properties as referred to a representative tetrachiral geometry.
Article
The rapid progress of advanced manufacturing, multidisciplinary integration and artificial intelligence has ushered in a new era of technological development in the design of lightweight, well-integrated, multifunctional, intelligent, flexible and biomimetic materials and structures. The traditional approach in structural research poses several intrinsic limitations on the practical performance of devices and instruments in harsh industrial environments, due to factors such as the disconnection between structural design and manufacturing, low efficiency in the manufacture of complex structures, reduced actual mechanical integrity and reliability of manufactured structures compared to the theoretical values obtained from structural design, insufficient level of multifunctional structural integration, and excessive economic cost. In addition, the advanced materials and structures incorporated in industrial equipment often need to withstand extreme service environments, and it is increasingly important to further integrate the design, manufacture, function, performance evaluation and industrial application of advanced structures, to provide the theoretical and technical bases for optimizing their fabrication. In view of the above, the authors propose a new research paradigm of “mechanostructures,” which aims to achieve target mechanical responses of structures, devices and equipment in extreme service environments by integrating their structural design, manufacturing and performance evaluation. By designing novel structures based on desired static and dynamic mechanical responses and considering the mechanical behavior throughout the whole deformation process, the new field of “mechanostructures” pursues an application-oriented structural design approach. As a typical example of mechanostructures, lightweight multifunctional lattice structures with high stiffness, strength, impact resistance, energy absorption capacity, shock wave attenuation and noise reduction show great potential for applications in aerospace, transportation, defense, biomedical, energy, machinery, equipment and other industrial fields. In this respect, the mechanical design of lattice metastructures inspired by polycrystalline microstructures is presented, starting with a discussion on typical mechanical properties and multifunctional performance conflicts, and demonstrating the scientific merits of “mechanostructures” based on the innovative structural design, manipulation of the multifunctional mechanical properties, and elaboration of the underlying physical mechanisms.
Article
This work aims to make an initial attempt to the controllable fracture performances of sheet lattices. Four types of variable-thickness sheet lattices (VTSLs) are designed using the Schwarz-P type triply periodic minimal surface, taking into account the manufacturability of the laser powder bed fusion technique. Three-point bending tests are conducted to investigate the fracture performance of VTSLs. Using ductile damage simulations, the fracture evolution process is analyzed, and mechanisms of fracture initiation and crack interaction are described. Experimental and numerical studies have established that distributions of local materials have a significant impact on fracture patterns, including intracellular and intercellular fractures. Under the same condition, the flexural strength, energy absorption capability, and fracture force can be improved for VTSLs with smooth transitions of neighboring unit cells. Specifically, the strength of VTSLs can be further enhanced when the homogenized material model is used. This work establishes a link between the controllable fracture performance and the additive manufacturing design of VTSLs.
Article
Cellular materials with smooth profiles, improved structural strength, and reduced elastic-anisotropy are eternal pursuit in bone-implant filed. However, it is a huge task to meet so many requirements. In this work, a novel class of mechanical metamaterials, named as crystal sheet lattices, were proposed. The elastic performances were investigated using representative elementary volume model and examined by quasi-static compression tests. The plastic performances and energy absorption behaviors were experimentally calibrated. Meanwhile, elastoplastic simulations were adopted to study the deformation mode on the structural strengthen mechanism. The results demonstrate that the reduced elastic-anisotropy can be achieved without complicated regulation process. Under the same material volume fraction, the stiffness, yield strength, and energy absorption capability were respectively increased about 30%–60%, 30%–150%, and 70%–280%, for most of crystal sheet lattices in comparison with their truss-based counterparts. Being open type cellular materials, the crystal sheet lattices are of high mass-specific mechanical performances. Due to the smooth profiles, large surface volume ratios, and enhanced mechanical performances, CSLs also have potentials to be utilized in lightweight and heat transportation fields.
Article
Lattice structures’ performance is directly determined by the topological structure and arrangement of cells. Here, we propose a lattice structure design approach that generates predictable mechanical properties by adjusting the three-dimensional posture and mirror arrays of cells. Specifically, the Euler angle (α β γ) is introduced to accurately control the three rotational degrees of freedom of the orthogonal unit cell. Then, a new unit cell with symmetric characteristics composed of multiple primitive rotation units is constructed by mirror operation, which is called super variable pose Octahedral Structures (SVPOS). The mathematical model of SVPOS topological structure is established, the mapping relationship between Euler Angle and relative density is deduced, and the mechanical response of structures with different Euler angles is studied. The results show that Structure of α = β = γ = 30˚ obtained the best performance, and the elastic modulus, compressive strength, and energy absorption at 50% strain of it are 553.1%, 471.9% and 529% of that of the octahedral, respectively. The new lattice structure can be obtained by adjusting the Euler Angle, its relative density and performance can be better predicted, which can provide experimental and theoretical basis for the design of lattice structures that meet different requirements.
Article
Additively manufactured lattice structures enable the design of tissue scaffolds with tailored mechanical properties, which can be implemented in porous biomaterials. The adaptation of bone to physiological loads results in anisotropic bone tissue properties which are optimized for site-specific loads; therefore, some bone sites are stiffer and stronger along the principal load direction compared to other orientations. In this work, a semi-analytical model was developed for the design of transversely isotropic lattice structures that can mimic the anisotropy characteristics of different types of bone tissue. Several design possibilities were explored, and a particular unit cell, which was best suited for additive manufacturing was further analyzed. The design of the unit cell was parameterized and in-silico analysis was performed via Finite Element Analysis. The structures were manufactured additively in metal and tested under compressive loads in different orientations. Finite element analysis showed good correlation with the semi-analytical model, especially for elastic constants with low relative densities. The anisotropy measured experimentally showed a variable accuracy, highlighting the deviations from designs to additively manufactured parts. Overall, the proposed model enables to exploit the anisotropy of lattice structures to design lighter scaffolds with higher porosity and increased permeability by aligning the scaffold with the principal direction of the load.
Article
In this paper, an analytical homogenization formula is derived based on the numerical representative volume element (RVE) method and asymptotic homogenization (AH) method for the effective elastic properties of truss lattice materials. Compared with the numerical homogenization method, the analytical homogenization formula can treat the geometric parameters of the truss structure, the elastic properties of the base material as well as the loading direction as independent variables without repeated modelling. Compared with the Gibson and Ashby’s meso-mechanics model, the analytical homogenization formula can not only obtain the same calculation results, but also can conveniently evaluate and control the elastic properties of the stretching-dominated compound truss lattice structure with complex truss configuration and multi-material composition. According to the anisotropy and deformation characteristics of the elementary trusses, design methodologies for stretching-dominated compound truss lattice structures with controllable anisotropy and optimal isotropic elasticity are proposed. Based on the analytical homogenization formula, the elastic isotropy conditions of the stretching-dominated compound truss lattice materials can be obtained. In the end, the designed 3D elastically isotropic truss lattice structures and 2D bi-material elastically isotropic lattice structures are manufactured and tested to verify the correctness of the isotropy conditions and calculation results obtained by the analytical homogenization formula.
Article
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The lattice structures are a particular type of structures made by the repetition of a unit cell and show great design opportunities. In addition, their structure is really close to some physiological tissues, which can allow their use to develop prostheses needed to the rehabilitation or replacement of a body part. However, their use is still limited, mainly due to the lack of methods to fully implement them during the production and to virtually predict all their mechanical properties. This problem is mostly caused by the computational effort and number of design parameters that the implementation of these materials in a Finite Element Modelling (FEM) analysis requires. Moreover, many common CAD software have a lack of materials libraries and geometry flexibility. In this work, samples with different lattice structures were manufactured by Laser Powder Bed Fusion technique using Co-Cr-Mo alloy. Compression tests were carried out to characterize their mechanical behavior. Subsequently, modelling and FE simulations were carried out to predict their mechanical response. In fact, a Finite Element Analysis allows to have a preview of final designed structure and to reduce the experimental tests otherwise needed to reach the final design, saving time and resources. Numerical simulations of the compression test were performed by FEM code Abaqus, in order to explore the possibilities and limitations of this approach for the study of lattice structures. Results of numerical simulations were compared with experimental data. Finally, NTopology software was also used to study the stiffness of the lattice structures according to the geometry of the investigated unit cells.
Article
Lattice structures with triply periodic minimal surfaces (TPMS) built using flexible materials are soft porous solids applicable in various fields, including biomedicine and tissue engineering. Such structures are also relevant for material extrusion additive manufacturing (MEAM), whose wide diffusion is pivotal to fostering their spread. Although design approaches are available to exploit the potential of soft TPMS, there are still manufacturing constraints that lead to practical limits on the shape and size of the structures that can be produced due to the complexities related to printing flexible materials. Besides, the computational models investigating the effect of cell type, the surface-to-volume fraction, and the combination of different periodic surfaces (i.e., graded or hybrid) on the mechanical behavior of these lattices are design aspects still debated. Here, the capabilities of MEAM to produce tailored soft lattice structures are explored by combining a design tool, numerical analyses, and mechanical testing using thermoplastic polyurethane (TPU) as feedstock material. The study addresses design issues, delves into optimum printing parameters, and analyzes a set of numerical parameters, which can be used for designing specific structures with tunable mechanical behavior, useful for healthcare and bioengineering. The printing parameters of three lattices, i.e., schwartz-P, gyroid, and honeycomb, with unit cell sizes spanning from 3 to 12 mm were studied. Their mechanical behavior was investigated using FEM simulations and mechanical testing. Lastly, the printability of graded and hybrid lattices with enhanced bearing-load capabilities have been demonstrated. Altogether, our findings addressed multiple challenges associated with developing soft lattice scaffolds with MEAM that can be used to fabricate innovative-engineered materials with tunable properties.
Article
Lattice-based mechanical metamaterials can be tailored for a wide variety of applications by modifying the underlying mesostructure. However, most existing lattice patterns take symmetry as a starting point. We show that asymmetric lattice patterns can be more likely to have certain mechanical properties than symmetric lattice patterns. To directly compare the effects of asymmetric versus symmetric lattice arrangements, a constrained design space is defined. A generative design process is used to generate both symmetric and asymmetric lattice patterns within the design space. Asymmetric lattice patterns are shown to have distinct metamaterial property spaces from symmetric lattice patterns. Key design features are identified that are present predominantly in asymmetric lattice patterns. We show that asymmetric lattice patterns with two of these features (arrows and spider nodes) are more likely to induce a broader range of Poisson’s ratios and larger shear stiffness values, respectively, compared to lattice patterns without these features. In addition, we show that symmetry can play a role in hampering the impact of multiple features when present. This work provides insights into the benefits of using asymmetric lattice patterns in select metamaterial design applications.
Article
Additive manufacturing techniques facilitate fabrication of polymeric lattice structures comprising complex cell architectures, via the ability to fabricate component struts at specified angles. However, the inherent layer-wise fabrication process, especially for Fused Deposition Modelling (FDM), introduces angle-dependent properties depending on the orientation with respect to the printing bed. This results in lattices displaying different mechanical responses, depending on the direction of loading. The present study examines how the static and dynamic tensile material properties of cell strut material are influenced by the angle of printing. Specimens printed at various angles were subjected to quasi-static tensile loading, as well as dynamic extension using a tensile split Hopkinson bar device. It was found that the degree of rate-sensitivity depends on the printing angle. Octet and Hybrid Structure (HS) lattices were also fabricated and subjected to quasi-static and impact compression along the printing direction and transverse to it. The results show that the load-deformation responses and lattice crushing patterns differ significantly. Finite element models, incorporating both printing angle and rate dependent strut material and failure properties, were established, to analyse the deformation with respect to loading direction, at both lattice sample and cell-component levels.
Article
Obtaining multifunctionality from microstructure instead of constituents provides a new direction for developing multifunctional materials. Periodic hollow sphere foam (HSF) offers one lightweight structural motif with integrated open and closed cells, high energy absorption, low thermal conductivity, snap-through instability, and triple-negative material indices. Here, we investigate the direction-dependent mechanical property, instability, and elastic wave isolation behavior of HSFs. Explicit formulas, stereographic projections, and general scaling relationships are developed to quantify and visualize the anisotropic mechanical properties of HSFs. By investigating the directional wave propagation in HSFs, extremely wide phononic band gaps are discovered in the HSFs. The derived formulas and the simulation-informed parametric maps allow us to design HSFs with desired static and dynamic anisotropic property profiles, including tailorable direction-dependent stiffness/shear modulus, negative Poisson’s ratio, and wave isolation properties. Building upon these results, multifunctional design concepts of HSFs are further set forth. This study not only reveals tailorable mechanical anisotropy and band gap in HSFs, but also developed a general approach to investigate the direction-dependent properties of periodic materials, enabling multifunctional applications where lightweight, direction-dependent property, wave attenuation, and programmability are required simultaneously.
Article
Nature materials usually possess unique hierarchical structures, like spongy bone, tendon and bamboo, and often exhibit remarkable mechanical properties. In this paper, inspired by the structural hierarchy of biological materials, the novel configuration design of unit cell with inner hierarchy was developed. The new lattice configuration takes advantage of the space filling and volume utilization of original BCC structure. The hierarchical lattices with 5×5×5 unit cells were manufactured by digital light processing (DLP) printing technique, using a hard-tough resin material. Numerical simulation and quasi-static experiment were performed to investigate the mechanical performance and deformation mechanisms of the lattice structures. The novel lattice configuration exhibits superior mechanical properties and enhanced energy absorption capacity with respect to conventional BCC lattice, e.g. when loading along x-axis, the improvement can be 38.9% for specific stiffness, 36.5% for specific energy absorption (SEA) and 73.1% for the crash load efficiency (CLE). Besides, the enhancement of mechanical performance and energy absorption capacity is more strong when loading along the z-axis. The mechanical interaction effect between structural hierarchy, e.g. master and slave cells, is proved to be the main reason that contributes to the enhancement of mechanical properties of hierarchical lattices. The designed novel configuration of hierarchical lattice will enrich the current lattice systems and promote the development of multifunctional applications in the future.
Article
The effective mechanical properties of plane fibre networks have been derived using discrete homogenization. The resulting equivalent continuum presents strong directional properties, with anisotropy ratios much larger than those that can be found in natural materials. The directionality of the response of the equivalent continuum increases with the slenderness of the fibres, leading to peculiar behaviours, like the formation of narrow bands where the strains present very large gradient. The results of the homogenization have been validated comparing the response of the continuum with the response of “discrete” models, for which each element is modelled as a slender beam, and boundary conditions are accounted for in an exact way.
Article
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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.
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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.
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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.
Article
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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.
Article
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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.
Article
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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.
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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.
Article
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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.
Book
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.
Chapter
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.
Book
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
Cellular solids include engineering honeycombs and foams (which can now be made from polymers, metals, ceramics, and composites) as well as natural materials, such as wood, cork, and cancellous bone. This new edition of a classic work details current understanding of the structure and mechanical behavior of cellular materials, and the ways in which they can be exploited in engineering design. Gibson and Ashby have brought the book completely up to date, including new work on processing of metallic and ceramic foams and on the mechanical, electrical and acoustic properties of cellular solids. Data for commercially available foams are presented on material property charts; two new case studies show how the charts are used for selection of foams in engineering design. Over 150 references appearing in the literature since the publication of the first edition are cited. It will be of interest to graduate students and researchers in materials science and engineering. © Lorna J. Gibson and Michael F. Ashby, 1988 and Lorna J. Gibson and Michael F. Ashby, 1997.
Article
This paper demonstrates a simple finite element implementation of Lagrange multipliers to model the mechanical behaviour of an orthotropic composite material. The research shows the proper set of kinematic boundary conditions that must be applied in 2D plane stress elasticity to achieve the correct unit strain vectors that, upon interrogation of the associated Lagrange multipliers, give the stresses induced by these strain vectors. From these stresses the terms in the elasticity matrix can be evaluated. As well as demonstrating the correct kinematic conditions required, the paper presents the consequences of applying intuitive but incorrect conditions. © 1997 John Wiley & Sons, Ltd.
Article
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.
Article
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%.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
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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.
Article
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.
Article
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.
  • M A Meyers
  • K K Chawla
M.A. Meyers, K.K. Chawla, Mechanical Behavior of Materials, Cambridge University Press, Cambridge, 2009.
  • T A Schaedler
  • A J Jacobsen
  • A Torrents
  • A E Sorensen