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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.

To read the full-text of this research,

you can request a copy directly from the authors.

... The quantity E=3ð1 À 2mÞ is called the modulus of volume expansion, i.e., the bulk modulus, K. As such, the compressibility of a solid, being the inverse of the bulk modulus [64], can be given by the derivatives [56], K ¼ À 1 V dV dp , where V is volume. These elasticity constants are positive and definite. ...

... Compressibility is the inverse of the bulk modulus K, and is a measure of the relative volume change of a solid or fluid as a response to a pressure change [64]. Hence, normally the compressibility is positive sign, only in the regime of strong ellipticity [67] it may be negative in few natural materials. ...

... A bi-directional evolutionary structural optimisation method [64] in topology is used to design materials, covering various properties such as stiffness. A statistical physics theory for negative compressibility transitions exists. ...

Mechanical metamaterials are man-made structures with counterintuitive mechanical properties that originate in the geometry of their unit cell instead of the properties of each component. The typical mechanical metamaterials are generally associated with the four elastic constants, the Young's modulus E, shear modulus G, bulk modulus K and Poisson's ratio υ the former three of which correspond to the stiffness, rigidity, and compressibility of a material from an engineering point of view. Here we review the important advancements in structural topology optimisation of the underlying design principles, coupled with experimental fabrication, thereby to obtain various counterintuitive mechanical properties. Further, a clear classification of mechanical metamaterials have been established based on the fundamental material mechanics. Consequently, mechanical metamaterials can be divide into strong-lightweight (E/ρ), pattern transformation with tunable stiffness, negative compressibility (−4G/3 < K < 0), Pentamode metamaterials (G ≪ K) and auxetic metamaterials (G ≫ K), simultaneously using topology optimisation to share various fancy but feasible mechanical properties, ultralight, ultra-stiffness, well-controllable stiffness, vanishing shear modulus, negative compressibility and negative Poisson's ratio. We provide here a broad overview of significant potential mechanical metamaterials together with the upcoming challenges in the intriguing and promising research field.

... Another extension of the BESO method for the design of orthotropic materials was carried out by Xie et al. [166], in which a series of designs of orthotropic materials with various magnitudes of negative/zero compressibility in one or two directions were provided. Compressibility is a measure of the relative volume change of a solid or fluid as a response to a pressure change. ...

... There has been increasing interest in the negative compressibility behavior, mostly due to its many potential applications such as sensitive pressure sensors, pressure driven actuator and optical telecommunication cables. Apart form providing numerical designs in [166], a physical prototype of one of such material designs is fabricated using a 3D printer and tested in the laboratory under either unidirectional loading or triaxial compression. Fig. 40 shows a typical negative linear compressibility (NLC) material design. ...

... A typical NLC material design and triaxial compression test on a correspondingly printed 8 × 8 × 8 cells[166] ...

The evolutionary structural optimization (ESO) method developed by Xie and Steven (1993, [162]), an important branch of topology optimization, has undergone tremendous development over the past decades. Among all its variants , the convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) method developed by Huang and Xie (2007, [48]) allowing both material removal and addition, has become a widely adopted design methodology for both academic research and engineering applications because of its efficiency and robustness. This paper intends to present a comprehensive review on the development of ESO-type methods, in particular the latest con-vergent and mesh-independent BESO method is highlighted. Recent applications of the BESO method to the design of advanced structures and materials are summarized. Compact Malab codes using the BESO method for benchmark structural and material microstructural designs are also provided.

... Furthermore, negative [64,71,73,74,[77][78][79][92][93][94][95][96][97][98][99][100][101] or zero [55] linear compressibility and negative [51,54,65,66,74,83,100] or zero [100] Poisson's ratio phenomena have been encountered for many of these materials. The negative-linear compressibility (NLC) [129][130][131], zero linear compressibility (ZLC) [132][133][134][135] and negative Poisson's ratio phenomena (NPR) [136][137][138][139] have multiple potential applications [129,131,[133][134][135][139][140][141][142][143][144][145][146][147][148], the most well-known being the development of ultrasensitive pressure sensors and actuators [129,139,140]. The relevance of the research on the behavior of highly porous materials under the effect of pressure in materials science has further increased since the application of high pressures to this type of materials has allowed for the design of new advanced functional materials, thus expanding the limits of conventional synthetic chemistry. ...

... However, the value of for ALPO-8 is the lowest linear compressibility found for all of the ALPO materials considered, ( = 1.81 TPa ). The criterium usually used for zero linear compressibility (ZLC) [102,103,[132][133][134][135] is that the absolute value of the linear compressibility along a certain direction is smaller than 1.0 TPa , | | 1.0 TPa [133]. ...

Here, a detailed mechanical characterization of five important anhydrous microporous aluminophosphate materials (VPI-5, ALPO-8, ALPO-5, ALPO-18, and ALPO-31) is performed using first principles methods based on periodic density functional theory. These materials are characterized by the presence of large empty structural channels expanding along several different crystallographic directions. The elasticity tensors, mechanical properties, and compressibility functions of these materials are determined and analyzed. All of these materials have a common elastic behavior and share many mechanical properties. They are largely incompressible at zero pressure, the compressibilities along the three crystallographic directions being frequently smaller than 5 . Notably, the compressibilities of ALPO-5 and ALPO-31 along the three principal directions are smaller than this threshold. Likewise, the compressibilities of ALPO-18 along two directions are smaller than 5 . All of the considered materials are shear resistant and ductile due to the large bulk to shear moduli ratio. Furthermore, all of these materials have very small mechanical anisotropies. ALPO-18 exhibits the negative linear compressibility (NLC) phenomenon for external pressures in the range P = 1.21 to P = 2.70 GPa. The minimum value of the compressibility along the [1 0 0] direction, 30.9 , is encountered for P = 2.04 GPa. The NLC effect in this material can be rationalized using the empty channel structural mechanism. The effect of water molecule adsorption in the channels of ALPO-18 is assessed by studying the hydrated ALPO-18 material (ALPO-18W). ALPO-18W is much more compressible and less ductile than ALPO-18 and does not present NLC effects. Finally, the effect of aging and pressure polymorphism in the mechanical properties of VPI-5 and ALPO-5 is studied. As hydration, aging leads to significant variations in the elastic properties of VPI-5 and increases substantially its compressibility. For ALPO-5, pressure polymorphism has a small impact in its elasticity at zero pressure but a large influence at high pressure.

... strains due to changes in temperature, environmental pressure, hygroscopic concentration and other environmental conditions-as much as possible so as to minimize the environmental stresses when the structures are constrained. Specifically, materials with zero or near zero thermal expansion [11][12][13], compressibility [14][15][16] and hygroscopic expansion [17][18][19] have been explored. However, experience in practical considerations has informed designers that it is (almost) impossible to design every part of a structure or device using materials with zero or near-zero environmental expansion coefficients. ...

... The combined influence from the hygroscopic strain of the central rods and the ratio of change in the hygroscopic concentration of the central rod to that of the surrounding environment ΔH c /ΔH env on the dimensionless effective coefficient of hygroscopic expansion α H eff =α H c is plotted in Fig. 5 (right) under the same special case using Eq. (14). ...

This paper proposes a truss microstructure that exhibits negative properties during increase in temperature or moisture and decrease in pressure but reverses to conventional properties during decrease in temperature or moisture and increase in pressure so that the material contracts regardless of the direction by which the environment condition changes. Both primary and secondary cells are bounded by 4 side rods to form squares at original state, with each primary cell containing a central rod connected diagonally. Sign-switchability of material properties is observed when the central rod is more responsive than the side rods during environmental fluctuations. The proposed microstructure exhibits zero environmental expansion when the environmental change is insignificant, and can be further designed to exhibit zero environmental expansion for large change in environmental fluctuation.
Graphical abstractThis metamaterial exhibits expansion coefficients that switch between positive and negative values such that it always contracts with environmental changes.

... Structural topology optimization technique is an effective approach to achieve the best structural performance with limited amount of materials [1][2][3] . To date, topology optimization has been used to solve one-scale design problems either for macrostructures to improve their structural performance or for materials to develop new microstructures with prescribed or extreme properties [4][5][6] . ...

... Series: Materials Science and Engineering 531 (2019) 012046 IOP Publishing doi:10.1088/1757-899X/531/1/012046 5 Young's modulus E0 = 1.0, Poisson's ratio μ=0.3. ...

An Integrated structural and material topology optimization method considering optimal material orientation is presented based the on bi-direction evolutionary structural optimization (BESO) method. It is assumed that the macrostructure is composed of uniform cellular material but with different orientation. The homogenization method is used to calculate the effective material properties which builds a connection between material and structure. The continuous material orientation design variables and the discrete topology design variables are treated hierarchically in an iteration. The principal stress method is adopted and embedded to determine the optimal material orientation, meanwhile the topologies of the macrostructure and its material microstructure are concurrently optimized by using the BESO method. Numerical examples are conducted to demonstrate the effectiveness of the proposed optimization algorithm.

... Simulation results are verified against experiments with soft lattices realized by PolyJet multi-material polymer 3D printing, highlighting the potential for simulation-driven, digital design and application of non-uniform and curved soft lattice structures. as negative Poisson's ratio effects [16][17][18]. ...

... The measurement data for the material characterization can be found in the Appendix. For all materials, a Poisson's ratio of = 0.45 and mass density = 1,150 kg/m 3 are used in the simulations [18,33]. Since the failure strains of these 3D printed materials are beyond 100% [33], we model them as elastic even for large structural strains of 20% and more. ...

Lattice structures are frequently found in nature and engineering due to their myriad attractive properties, with applications ranging from molecular to architectural scales. Lattices have also become a key concept in additive manufacturing, which enables precise fabrication of complex lattices that would not be possible otherwise. While design and simulation tools for stiff lattices are common, here we present a digital design and manufacturing approach for soft lattices structures subject to large deformations and instabilities, for which applications in soft robotics, healthcare, personal protection, energy absorption, fashion and design are rapidly emerging. Our framework admits soft lattices with curved members conforming to freeform geometries, and with variable, gradually changing member thickness and material, allowing the local control of stiffness. We model the lattice members as 3D curved rods and using a spline-based isogeometric method that allows the efficient simulation of nonlinear, large deformation behavior of these structures directly from the CAD geometries. Furthermore, we enhance the formulation with a new joint stiffening approach, which is based on parameters derived from the actual node geometries. Simulation results are verified against experiments with soft lattices realized by PolyJet multi-material polymer 3D printing, highlighting the potential for design and application of non-uniform and curved soft lattice structures.

... Meanwhile, topology optimization with inverse homogenization technique was proposed initially in microstructural design of porous and composite materials [19,20]. The increasing advancement of additive manufacturing technology makes it possible to fabricate various man-made materials, such as porous material with negative Poisson's ratio over large deformations [21], orthotropic material for negative or zero compressibility [22]. The stateof-the-art for material design via topology optimization can be referenced in Ref. [23]. ...

... By combining Eqs. (21), (22), and (26), the mathematical optimization model determined by Eq. (9) can be rewritten as ...

The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously. Nodal displacement of macrostructure and effective thermal conductivity of microstructure are referred as the constraint functions, which means taking into account both the load-carrying capabilities and the thermal insulation properties. The effective properties of porous material derived from numerical homogenization are used for macro-structural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are utilized for the sensitivity analysis of the microstructure. Design variables in form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.

... In the first design approach, a framework with NLC property was designed which the corresponding mechanism for NLC was similar to the wine-rack model. The topology design of the microstructure was created through the simplification of the results of our previous work [42] on the application of the bi-directional evolutionary structural optimization (BESO) to find the microstructure of metamaterial with negative compressibility. ...

... This research demonstrated the effectiveness of two proposed approaches to designing new composite structures with negative linear compressibility (NLC). The first approach was to simplify the topology optimization results from previous research [42] to generate the framework reinforcement for a new NLC composite structure. The second approach was to use the discrete truss elements of a special pattern to reinforce the crushable foam for generating a new type of NLC composite structure. ...

The phenomenon of negative linear compressibility has attracted much interest because of its unusual deformation features with many potential applications. However, the design and fabrication of materials and structures with negative linear compressibility are limited. In this paper, we proposed two approaches to designing and fabricating new composite structures with negative linear compressibility. The effectiveness of the proposed design approaches was validated experimentally by applying uniformly distributed pressure to all surfaces of bulk specimens. The deformation features, strain history, and the effective area reduction of the specimens were analyzed from the experimental data. The results clearly demonstrated the feasibility of the proposed designing and manufacturing approaches for realizing composites with negative linear compressibility.

... In addition to design based on mechanical theory [7,1,9], numerical optimization could be used for the design of microstructural shapes [10,5]. Topology optimization, which enables fundamental optimization of structures, including changes in topology (number of holes) [11,12], contributes to the design of materials with extreme effective physical properties [10,8,13,5,6,[14][15][16]. ...

... This could be a suitable fabrication method for porous or composite materials with optimized internal geometries. Several experimental verifications of the optimized internal geometries have been reported recently in elastic performances [27][28][29]16,30,31]. Electron beam melting could also be used in porous metal fabrication [32,33]. ...

Additive manufacturing may be a novel method for fabricating porous materials. These materials can achieve effective performance because of their internal geometries. Metal-additive manufacturing is expected to utilize thermal conduction materials and devices. We have developed a porous metal with effective isotropic thermal conductivity by using metal-selective laser melting additive manufacturing. The internal pore structure was designed by topology optimization, which is the most effective structural optimization technique to maximize effective thermal conductivity. The designed structure was converted to a three-dimensional STL model, which is a native digital format of additive manufacturing, and assembled as a test piece. Effective thermal conductivity was measured by a steady-state method in which the effective thermal conductivity was calculated from a one-dimensional temperature gradient and the heat flux of the test pieces. The test pieces showed an effective thermal conductivity close to the Hashin–Shtrikman or Maxwell–Eucken bound, which is the theoretical limit of effective performance with an error less than 10%.

... The asymptotic homogenization-based topology optimization approach is an efficient structural optimization technique for the design of lattice structures since the seminal work [8], and has become a mature tool in structural design with decades of developments. For microstructural design, the inverse homogenization approach [9] is commonly adopted for specified or extreme material properties, such as negative or zero compressibility design optimization [10], thermoelastic composites optimization for maximized stiffness and heat transfer [11], programmable Poisson's ratio over large deformation [12], etc. Microstructural optimization for metamaterial design based on the Floquet-Bloch theory has also been extensively investigated in wave propagation, and the conventional gradient-based optimizers has been widely adopted [13][14][15], yet the computational resources could be demanding. More recently, efforts has been made to improve the efficiency for optimization, such as design of innovative mechanical beam lattices via adaptive surrogate models and machine learning [16,17], and onedimensional acoustic metamaterials using machine learning and cell concatenation [18]. ...

This paper presents a novel two-step homogenization-based topology optimization and de-homogenization method for the design of graded lattice structures. The lattice orientation and material layout are first optimized for square base cells in the macro scale. Then by introducing the lattice stretching design variables of micro base cells, which bridge the base cell distortion with lattice stretch, the error residual of mapping functions is integrated with compliance formulation to form a novel mixed optimization formulation, concurrently optimizing structural performance and mapping functions. The advantage of this formulation is two-fold. First, the micro design space is relaxed from square base cells to rectangular ones so that performance improvement is further expected. Second, an excellent agreement, in both shape and performance, between the projected single-scale lattice structures with the homogenization results is secured, as compared to the frequently adopted post-process procedure of constructing single-scale lattices, where performance deviation could arise for specific microstructural patterns. With the optimized mapping functions, de-homogenization procedure is carried out to construct single-scale spatially graded lattice structures, where a simple filter-projection operation is proposed to obtain fine-scale smoothed boundaries from coarse-scale homogenization results with zig-zag boundaries. Several numerical examples are presented and compared with conventional post-process treatment results to show the validity of the proposed method, and different kinds of lattice patterns adopted in this work show its versatility for a broad range of lattice patterns.

... The homogenized properties of the metamaterials are highly dependent on their micro-architectures [3]. In recent years, many researches have been carried out on various mechanical metamaterials, including multistable metamaterial [9][10][11][12], bi-mode/pentamode metamaterial [13][14][15][16], auxetic metamaterial [17][18][19][20], chiral metamaterial [21][22][23][24], programmable metamaterial [25][26][27][28], and metamaterials with negative swelling [29,30], negative stiffness [31], and negative compressibility [32,33]. For developments in the field of mechanical metamaterials, interested readers are referred to the review papers [34,35]. ...

Poisson’s ratio is an important property defining the relationship between lateral and longitudinal deformations. While most conventional materials have positive Poisson’s ratios, auxetic metamaterials exhibit negative Poisson’s ratios and will contract laterally under vertical compression and expand laterally under vertical tension. In this study, we develop a new type of mechanical metamaterial, which always undergoes lateral expansion regardless of the sign of the uniaxial load. The unit cell configuration is presented by combining a re-entrant negative Poisson’s ratio structure and a hexagonal structure. A new stiffness- and deformation mode-switching mechanism is realized by exploiting contact nonlinearity in the unit cell configuration design. This novel metamaterial can exhibit positive, negative and zero Poisson’s ratios under different loading directions. Its mechanical properties are verified through numerical simulations and experimental tests. The unit cell configuration is also extended to 3D metamaterials and metastructures. This study demonstrates that certain desired mechanical properties can be achieved by introducing contact nonlinearity and deformation mode switching into metamaterial design.

... Metamaterials has penetrated into various disciplines, including electromagnetism [1], optics [2], acoustics [3], heat [4] and mechanics [5]. The emergence of mechanical metamaterials has brought about many innovative design concepts, such as auxetic materials [6][7][8], negative stiffness materials [9], negative compressibility materials [10], lightweight high strength materials [11] and super-fluids. As the most widely studied mechanical metamaterials, auxetic materials have many excellent properties and wide application prospects. ...

A novel type of tubular structure has been proposed in this paper, which is the first tubular structure with auxeticity in the wall thickness as well as in the radial direction. This tubular structure exhibits good stability under axial compression. The most innovative feature is that its inner diameter and outer diameter have opposite deformation directions. When axially compressed (stretched), its outer diameter will become smaller (larger) while its inner diameter will become larger (smaller). Besides, its closed surfaces of tube wall can broaden its applications in the fields of civil engineering and mechanical engineering. The accuracy of the finite element model was verified by comparison between experiments and numerical analysis. Deformation characteristics of tubular models generated by offset method and rotation method were studied. The influence of cell layers, PSF (Pattern Scale Factor) value and t/R (the ratio of wall thickness t to diameter R) value was also studied by parametric analysis.
(20) (PDF) A novel type of tubular structure with auxeticity both in radial direction and wall thickness. Available from: https://www.researchgate.net/publication/352400149_A_novel_type_of_tubular_structure_with_auxeticity_both_in_radial_direction_and_wall_thickness [accessed Jun 15 2021].

... Metamaterials has penetrated into various disciplines, including electromagnetism [1], optics [2], acoustics [3], heat [4] and mechanics [5]. The emergence of mechanical metamaterials has brought about many innovative design concepts, such as auxetic materials [6][7][8], negative stiffness materials [9], negative compressibility materials [10], lightweight high strength materials [11] and super-fluids. As the most widely studied mechanical metamaterials, auxetic materials have many excellent properties and wide application prospects. ...

A novel type of tubular structure has been proposed in this paper, which is the first tubular structure with auxeticity in the wall thickness as well as in the radial direction. This tubular structure exhibits good stability under axial compression. The most innovative feature is that its inner diameter and outer diameter have opposite deformation directions. When axially compressed (stretched), its outer diameter will become smaller (larger) while its inner diameter will become larger (smaller). Besides, its closed surfaces of tube wall can broaden its applications in the fields of civil engineering and mechanical engineering. The accuracy of the finite element model was verified by comparison between experiments and numerical analysis. Deformation characteristics of tubular models generated by offset method and rotation method were studied. The influence of cell layers, PSF (Pattern Scale Factor) value and t/R (the ratio of wall thickness t to diameter R) value was also studied by parametric analysis.

... The NPR material was originally named "auxetics" (derived from the Greek word "auxetikos") by Evans in1991 [1], which means "that which tends to increase". Compared to traditional materials with positive Poisson's ratio, auxetic metamaterials have the following improved mechanical properties: (a) in-plane indentation resistance [2,3]; (b) shear resistance [4][5][6]; (c) synclastic behaviour [7,8]; (d) energy absorption [9][10][11][12][13][14][15]; (e) fracture toughness [16][17][18]; (f) negative compliance [19][20][21] and, (g) sound insulation [22][23][24]. ...

Auxetic materials and structures have attracted increasing attention because of their extraordinary mechanical properties. Various types of auxetic tubular structures have been designed and studied in diverse fields, including mechanical and medical engineering. In this paper, design methods and advanced manufacturing technologies of auxetic tubular structures are extensively reviewed, including various types of cellular auxetic tubes, nonporous and porous auxetic tubes, macro and micro auxetic tubes. Furthermore, auxetic behaviour, mechanical properties and potential applications of auxetic tubular structures are elaborated. Finally, the challenges and opportunities on the auxetic tubes are discussed to inspire future research work.

... Topology optimization methods of continuum structures, e.g. density-based method, level set method and evolutionary structural optimization method, have been applied to design metamaterials [25][26][27][28]. For discrete structures, the most popular topology optimization method should be the ground structure method, which has also been applied in design optimization of mechanical metamaterials [29][30][31]. ...

Pentamode metamaterials are a new class of artificially engineering three-dimensional lattice composites. There exist a few types of pentamode metamaterials that are dominated by ad hoc design motifs, while a systematic design approach is still missing. This paper will present an efficient topological optimization methodology to discover a series of novel pentamode lattice microarchitectures over a range of effective material properties. Firstly, the necessary and sufficient condition that is required for elasticity constants of pentamode micro lattices with at least elastically orthotropic symmetry is derived. Secondly, a general mathematical formulation for design optimization of such pentamode micro lattices is developed. Thirdly, a truss-based three-dimensional ground structure with geometrically orthotropic symmetry is generated, with geometric constraints to avoid intersection and overlap of truss bars within the ground structure. The genetic algorithm is then used to solve the topology optimization problem described by the ground structure. Finally, twenty-four pentamode lattices are designed to demonstrate the effectiveness of the proposed method.

... 1.66(11)/TPa, and 25.63(97)/TPa, respectively (The transformation matrix between crystallographic and mechanical principal axes see Table S3). Note that the compressibility along the Y-axis is much smaller than that in the majority of materials and can be categorized into ZLC 25,26 . Therefore, the NLC, ZLC, and PLC are simultaneously presented in LiBO 2 . ...

Anomalous mechanical materials, with counterintuitive stress-strain responding behaviors, have emerged as novel type of functional materials with highly enhanced performances. Here we demonstrate that the materials with coexisting negative, zero and positive linear compressibilities can squeeze three-dimensional volume compressibility into one dimension, and provide a general and effective way to precisely stabilize the transmission processes under high pressure. We propose a “corrugated-graphite-like” structural model and discover lithium metaborate (LiBO2) to be the first material with such a mechanical behavior. The capability to keep the flux density stability under pressure in LiBO2 is at least two orders higher than that in conventional materials. Our study opens a way to the design and search of ultrastable transmission materials under extreme conditions.

... Mechanical metamaterials have attracted significant attention due to their extraordinary mechanical characteristics resulted in complex designs of microstructures [329][330][331]. To manoeuvre the mechanical properties and obtain desirable performance, it is necessary to establish the relations between the dominant parameters of the local microstructures and the mechanical performance of the global metamaterials [332,333]. ...

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

... Recently there has been increasing interest in the fabrication and application of soft lattices and metamaterials which can achieve large elastic deformations [11,12], exploit mechanical instabilities and buckling [13,14], recoverably absorb energy and mitigate vibrations [15], exhibit auxetic behaviour [16,17,18], have thermally tuneable and self-healing properties [19], or shape-memory and shape-morphing abilities [20,21]. Mechanical modelling of such soft lattice structures involves large deformations and finite strains and in particular recoverable elastic buckling of some struts followed by their post-buckling and bending behaviour [22]. ...

Soft lattice structures and beam-metamaterials made of hyperelastic, rubbery materials undergo large elastic deformations and exhibit structural instabilities in the form of micro-buckling of struts under both compression and tension. In this work, the large-deformation nonlinear elastic behaviour of beam-lattice metamaterials is investigated by micromechanical nonlinear buckling analysis. The micromechanical 3D beam finite element model uses a primary linear buckling analysis to incorporate the effect of geometric imperfections into a subsequent nonlinear post-buckling analysis. The micromechanical computational model is validated against tensile and compressive experiments on a 3D-printed sample lattice structure manufactured via multi-material jetting. For the development and calibration of macroscale continuum constitutive models for nonlinear elastic deformation of soft lattice structures at finite strains, virtual characterization tests are conducted to quantify the effective nonlinear response of representative unit cells under periodic boundary conditions. These standard tests, commonly used for hyperelastic material characterization, include uniaxial, biaxial, planar and volumetric tension and compression, as well as simple shear. It is observed that besides the well-known stretch- and bending-dominated behaviour of cellular structures, some lattice types are dominated by buckling and post-buckling response. For multiscale simulation based on nonlinear homogenization, the uniaxial standard test results are used to derive parametric hyperelastic constitutive relations for the effective constitutive behaviour of representative unit cells in terms of lattice aspect ratio. Finally, a comparative study for compressive deformation of a sample sandwich lattice structure simulated by both full-scale beam and continuum finite element models shows the feasibility and computational efficiency of the effective continuum model.

... Yang). been designed in the past few decades [19][20][21][22] . Thus, it is meaningful and feasible to combine the designs of metamaterials and the cores of sandwich structures. ...

... Referring to many applications of TO [van Dijk, Maute, Langelaar et al. (2013)], the design for engineering cellular materials attracted increasing attentions. Since the inverse homogenization method [Sigmund (1994)] has been adopted to predict effective properties of cellular microstructures, the design of microstructures is able to be formulated as an inverse problem for yielding the expected macro performance [Andreassen, Lazarov and Sigmund (2014); Takezawa, Kobashi and Kitamura (2015)] and prescribed properties [Xie, Yang, Shen et al. (2014)]. An extended design strategy of optimizing the functionally graded microstructures (GMs) is considered as alternative to offer more mechanical advantages for external stimuli. ...

This paper proposes a multiscale isogeometric topology optimization (ITO) method where the configuration and layout of microstructures are optimized simultaneously. At micro scale, a shape deformation method is presented to transform a prototype microstructure (PM) for obtaining a series of graded microstructures (GMs), where microstructural skeleton based on the level set framework is applied to retain more topology features and improve the connectability. For the macro scale calculation, the effective mechanical properties can be estimated by means of the numerical homogenization method. By adopting identical non-uniform rational basis splines (NURBS) as basis functions for both parameterized level set model and isogeometric calculation model, the isogeometric analysis (IGA) is integrated into the level set method, which contributes to improving the accuracy and efficiency. Numerical examples demonstrate that, the proposed method is effective in improving the performance and manufacturability.

... On the other hand, microstructural design is a technique for the distribution of material, at a smaller scale, to optimize material properties. Through microstructural design, one can customize various material behavior (Osanov and Guest 2016) including bulk/shear modulus (Huang et al. 2011), Poisson's ratio (Vogiatzis et al. 2017;Xie et al. 2014), thermal expansion (Sigmund and Torquato 1997), elasticity tensor (Sigmund 1994), and other extremal properties (Sigmund 2000). For example, Fig. 1b illustrates an optimal microstructure, once again computed via SIMP, for maximizing shear modulus. ...

Microstructural topology optimization (MTO) is the simultaneous optimization of macroscale topology and microscale structure. MTO holds the promise of enhancing product-performance beyond what is possible today. Furthermore, with the advent of additive manufacturing, the resulting multiscale structures can be fabricated with relative ease. There are however two significant challenges associated with MTO: (1) high computational cost, and (2) potential loss of microstructural connectivity. In this paper, a novel density-and-strain-based K-means clustering method is proposed to reduce the computational cost of MTO. Further, a rotational degree of freedom is introduced to fully utilize the anisotropic nature of microstructures. Finally, the connectivity issue is addressed through auxiliary finite element fields. The proposed concepts are illustrated through several numerical examples applied to two-dimensional single-load problems.

... This allows observation of local features in the strain field caused by alterations to the print pattern [29]. This is important as it is well known that FFF components can have extreme anisotropies in their physical characteristics determined by print patterns [30,31]. ...

This work investigates the evolution of the tensile and structural properties of fused filament fabrication (FFF), formed polymers under gamma irradiation. Commercial off-the-shelf print filaments of Poly(lactic acid) (PLA), Thermoplastic polyurethane (TPU), Chlorinated polyethylene elastomer (CPE), Nylon, Acrylonitrile butadiene styrene (ABS) and Polycarbonate (PC) were exposed to gamma-ray doses of up to 5.3 MGy. The suitability of FFF-formed components made from these materials for use in radiation environments is evaluated by considering their structural properties. We identify clear trends in the structural properties of all the materials tested and correlate them with changes in the chemical structure. We find that Nylon shows the best performance under these conditions, with no change in ultimate tensile strength and an increase in stiffness. However, some of our findings suggest that the effect of additives to this type of filament may result in potentially undesirable adhesive properties. The organic polymer PLA was notably more radiation-sensitive than the other materials tested, showing 50% decrease in Young’s Modulus and ultimate tensile strength at order of magnitude lower radiation dose. A mechanism is proposed whereby FFF-processed components would have substantially different radiation tolerances than bulk material.

... Initially restricted to optimizing the geometry of structures, the technique has been extended to optimizing the topology of the phase within materials, e.g. in periodic microstructures, to design high performance materials [12][13][14][15][16][17][18][19][20] or materials with properties not found in nature (negative Poisson's ratio, zero compressibility, negative bulk modulus, etc. (see [21][22][23][24][25]) or complex multiphysics problems [26,27]. These techniques are based on optimizing the homogenized properties of the representative volume element, and using numerical solving methods like finite element to compute the homogenized properties [28], given one geometry of the phases and their microscopic properties. ...

We present a topology optimization for lattice structures in the case of non-separated scales, i.e. when the characteristic dimensions of the periodic unit cells in the lattice are not much smaller than the dimensions of the whole structure. The present method uses a coarse mesh corresponding to a homogenized medium taking into strain gradient through a non-local numerical homogenization method. Then, the topological optimization procedure only uses the values at the nodes of the coarse mesh, reducing drastically the computational times. We show that taking into account the strain gradient within the topological optimization procedure brings significant increase in the resulting stiffness of the optimized lattice structure when scales are not separated, as compared to using a homogenized model based on the scale separation assumption.

... Even rarer is zero linear/ area compressibility (ZLC/ZAC), with which property the material neither expands nor contracts in one/two specific direction(s) over a certain pressure range. 10 The excellent property endows ZLC/ZAC materials with potential applications such as in sonar, hydrophones, and telecommunication optical fibers that can keep their dimensions unchanged under high-pressure conditions such as the deep-sea environments. 11,12 ZLC/ZAC nanocomposites could be achieved by combining the NLC/NAC materials with materials with positive linear compressibility (PLC). ...

Materials with zero area compressibility (ZAC) can keep their crystal uncompressed in two specific directions upon uniformly compression. High-pressure angle-dispersive X-ray powder diffraction (ADXRD) experiments reveal a ZAC phenomenon in the ab-plane in crystal of a formate-based perovskite, [C(NH2)3][Cd(HCOO)3]. The ZAC behavior is ascribed to the unique rhombohedral [Cd(HCOO)3]⁻ frameworks and confirmed by density functional theory (DFT) calculations. For the first time, a near ZAC single material is explicitly report. This study opens up an exciting research field on pressure-resistant materials. We anticipate more ZAC materials to be discovered in the following explorations under the inspiration of this work.

... Unlike homogeneous natural materials, metamaterials are products of human ingenuity and their properties depend largely on internal construction rather than on parent materials [1]. Over the past few decades, programmed by suitable topology and configuration, these artificial materials have been studied and designed to obtain unconventional characteristics, such as 2D [2,3] and 3D [4][5][6] behaviors of negative or zero Poisson's ratio, negative or zero compressibility [7][8][9], tunable magnitude and prescribed directionality of thermal expansion [10,11], negative effective mass density and negative effective elastic modulus [12] and low-frequency sound absorption [13]. ...

Unidirectional, bidirectional and tridirectional Buckling-based Negative Stiffness (BNS) lattice metamaterials are designed by adding prefabricated curved beams into multidimensional rigid frames. Finite Element Analysis models are built, and their mechanical performance is investigated and discussed. First, geometric parameters of the curved beam were systematically studied with numerical analyses and the results were validated by theoretical solutions. Next, within unidirectional designs of different layer numbers, the basic properties of multilayer BNS metamaterials were revealed via quasi-static compressions. Then, the bidirectional and tridirectional designs were loaded on orthogonal axes to research both the quasi-static and dynamic behaviors. For dynamic analysis conditions, simulation scenarios of different impact velocities were implemented and compared. The results demonstrate that the proposed numerical analysis step has accurately predicted the force-displacement relations of both the curved beam and multilayer designs and the relations can be tuned via different geometric parameters. Moreover, the macroscopic performance of the metamaterials is sensitive to the rigidity of supporting frames. The shock force during impact is reduced down below the buckling thresholds of metamaterial designs and sharp impact damage is avoided. The presented metamaterials are able to undergo multiaxial stress conditions while retaining the negative stiffness effect and energy-absorbing nature and possess abundant freedom of parametric design, which is potentially useful in shock and vibration engineering.

... Obecna technologia pozwala na wydruk elementów o ortotropowych makroskopowych własnościach materiałowych [5,8,10,13] w których osnowa z tworzywa sztucznego zawiera dodatkowe wzmocnienia w postaci włókien szklanych lub węglowych. Inną z metod uzyskania ortotropii materiału jest modyfikacja jego mikrostruktury poprzez którą dodawane są puste przestrzenie w jego wnętrzu [7,22]. Ta technologia, połączona z modelowaniem komputerowym na przykład przy pomocy MES oraz metodami optymalizacji kształtu, pozwala zaprojektować strukturę materiału o założonych parametrach. ...

The article is describing the whole modeling process of viola da gamba soundboard. The most frequently used material for soundboards is spruce. It has been replaced by a composite material that enables using the 3D printing technology. Different material parameters require also different values of the soundboard parameters to maintain the characteristic timbre of the pattern instrument. For this purpose, a multi-objective optimization has been made which based on the computer model of soundboard created using FEM. A computer experiment was planned and carried out using an optimal space filling algorithm (OSF) in space for the three input parameters (soundboard thickness at the edge and center, side plate thickness), and four output parameters (two modal frequencies and corresponding MAC values). They have been taken into account within the cost function. Response surfaces has been determined by the universal kriging method. The optimization process of model parameters, realized by screening method, allowed to choose their values describing shape definitely thicker then wooden plate. The designed plate of viola da gamba has two modal frequencies with difference less than 9 % from pattern. Moreover, a MAC coefficient describing similarity of modes is higher than 0,7.

... [27,28]. The method has been applied to a wide range of problems like nonlinear structures [29], natural frequency maximization [30,31], material optimization and multiscale problems [32][33][34][35][36][37], multiphysics problems [38,39], etc. The design variables are updated based on the thresholds of sensitivity numbers corresponding to the objective function, and the thresholds are set based on the evolutionary ratios. ...

This work proposes an improved method for gradient-based topology optimization in a discrete setting of design variables. The method combines the features of BESO developed by Huang and Xie [1] and the discrete topology optimization method of Svanberg and Werme [2] to improve the effectiveness of binary variable optimization. Herein the objective and constraint functions are sequentially linearized using Taylor's first order approximation, similarly as carried out in [2]. Integer Linear Programming (ILP) is used to compute globally optimal solutions for these linear optimization problems, allowing the method to accommodate any type of constraints explicitly, without the need for any Lagrange multipliers or thresholds for sensitivities (like the modern BESO [1]), or heuristics (like the early ESO/BESO methods [3]). In the linearized problems, the constraint targets are relaxed so as to allow only small changes in topology during an update and to ensure the existence of feasible solutions for the ILP. This process of relaxing the constraints and updating the design variables by using ILP is repeated until convergence. The proposed method does not require any gradual refinement of mesh, unlike in [2] and the sensitivities every iteration are smoothened by using the mesh-independent BESO filter. Few examples of compliance minimization are shown to demonstrate that mathematical programming yields similar results as that of BESO for volume-constrained problems. Some examples of volume minimization subject to a compliance constraint are presented to demonstrate the effectiveness of the method in dealing with a non-volume constraint. Volume minimization with a compliance constraint in the case of design-dependent fluid pressure loading is also presented using the proposed method. An example is presented to show the effectiveness of the method in dealing with displacement constraints. The results signify that the method can be used for topology optimization problems involving non-volume constraints without the use of heuristics, Lagrange multipliers and hierarchical mesh refinement.

... Because of the uncommon feature which is equipped by auxetics, these auxetic materials and structures are superior to conventional materials and structures in terms of indentation resistance [163,231], shear resistance [167], synclastic behaviour [9], enhanced resilience [9], energy absorption [67,[232][233][234], fracture toughness [172], vibration control [235] and negative compliance [236][237][238]. ...

... In general, these methods can be considered gradient-based methods that rely on discrete design updates which result only in 0/1 solutions during the optimization process. Further works proved that the BESO method could efficiently solve different topology optimization problems such as for nonlinear structures [24], multiple materials design [25], natural frequencies maximization [28], self-weight loads [27] and the recents BESO applications on multiphysics [41,52,43] and multiscale [56,62,47,29,53] design problems. ...

This work presents an extended bi-directional evolutionary structural optimization (BESO) method applied to static structural design problems considering the interaction between viscous fluid flows and linearly elastic structures. The fluid flow is governed by incompressible and steady-state Navier-Stokes equations. Both domains are solved with the finite element method and simplifying conditions are assumed for the fluid-structure coupling, such as small structural displacements and deformations in a staggered method. The presented BESO method aims to minimize structural compliance in a so called “wet” optimization problem, in which the fluid loads location, direction and magnitude depend on the structural layout. In this type of design-dependent loading problem, density-based topology optimization methods require extra numerical techniques (usually mixed models with overlapping domains) in order to model the interaction of different governing equations during the optimization procedures. In this work, the discrete nature of the evolutionary topology optimization approach allows the fluid-structure boundaries to be modelled and modified straightforwardly by switching the discrete design variables between fluid and structural finite elements. Therefore, separate domains are used in this approach. Numerical results show that the BESO-based methods can be applied to this kind of multiphysics problem effectively and efficiently.

... More substantial advantages can be foreseen with the development of printing entirely new polymers and polymer composites. In a number of cases, the flexibility available in the design and manufacture of products allows new metamaterials to be created that have, by means of a specific microstructure, for example, improved or unusual mechanical [19], electrical [20] or magnetic [21] properties. Improved antennae can be made using optimized composites of dielectric and magnetic particles in polymers [22]. ...

Additive manufacturing (or 3D printing) opens the possibility of creating new designs and manufacturing objects with new materials rapidly and economically. Particularly for use with polymers and polymer composites, simple printers can make high quality products, and these can be produced easily in offices, schools and in workshops and laboratories. This technology has opened a route for many to test ideas or to make custom devices. It is possible to easily manufacture complex geometries that would be difficult or even impossible to create with traditional methods. Naturally this technology has attracted attention in many fields that include the production of medical devices and prostheses, mechanical engineering as well as basic sciences. Materials that are highly problematic to machine can be used. We illustrate process developments with an account of the production of printer parts to cope with polymer fillers that are hard and abrasive; new nozzles with ruby inserts designed for such materials are durable and can be used to print boron carbide composites. As with other materials, complex parts can be printed using boron carbide composites with fine structures, such as screw threads and labels to identify materials. General ideas about design for this new era of manufacturing customised parts are presented.

... Early research had been more focused on achieving extruded 2D metamaterial designs with tailored effective properties. More recently, 3D metamaterial designs have been achieved by directly coupling TO with AM, which include negative Poisson's ratio structures [26][27][28], materials with extreme compressibility [29], and multi-material designs [28,30]. Depending on the TO method used, most optimization results need post-processing work before they are suitable for AM. ...

Topology optimization has been widely studied and implemented as a powerful conceptual design tool in various engineering applications. However, the result from topology optimization has posed an implementation challenge to engineers due to the complexity of converting obtained solution into CAD data and then fabricating it into real parts. Over the past few years, the advanced additive manufacturing technology with new materials and higher resolution output capabilities has opened numerous opportunities to fill the gap between topology optimization and product application. In this study, an engineering procedure is presented for the conversion of topology optimization result to ready-to-print model for additive manufacturing. The steps of post-optimization handling are outlined, and the potential practical issues for the additive manufacturing implementation are discussed. A vehicle example for full frontal impact load path development by topology optimization with inertia relief approach is used to exhibit the employed additive manufacturing implementation process with a reduced scale part build. The arising implementation issues and needs are examined for future advance of topology optimization and additive manufacturing integration development.

... More recently, Xie et. al [22] studied the compressibility of negative or zero Poisson's ratio porous materials. Despite these early attempts, it is not completely clear that how Poisson's ratio could affect crush behavior quantitatively and what is an optimum in a range from conventional values to auxetic level [23]. ...

This paper treats the influence of auxetic foam on the crush response and energy absorption response of square-section tubes when subjected to uniaxial quasi-static loading. The study aims at quantifying the energy absorption capability of auxetic foam-filled square tubes for variations in wall thickness, initial height, aspect ratio and slenderness ratio of the tube. The capability of simulating the crush response of auxetic foam-filled tubes using the validated numerical models is also presented. Based on the experimental results, the influence of the auxetic foam in the thin-walled square tubes was quantified in terms of energy absorption capacity, specific energy absorption and crush force efficiency. It is evident that a thicker tube filled with auxetic foam is preferable if the energy absorption level is the primary goal, yet this compromises the crush force efficiency. The outcome of this present study is the establishment of empirical models for estimating the quasi-static crushing response of auxetic foam-filled tubes with varying slenderness ratio and aspect ratio.

The use of computational evolutionary strategies in the design of metamaterials with desired thermal expansion coefficients is uncommon due to the discrete nature of the design variables. This work presents a Bi-directional Evolutionary Structural Optimization (BESO) based methodology for designing orthotropic metamaterials with a specific thermal expansion coefficient using an objective function considering only the thermal expansion coefficients, with no constraints on geometry or stiffness. Topologies of the metamaterials, composed of two material phases and a void, are obtained using a material interpolation between neighboring material phases and three easy-to-implement numerical strategies to stabilize the evolutionary process. Two are on the sensitivity calculation and one is on the addition ratio’s value. The strategies applied to the sensitivity numbers are proposed to avoid the positive and negative values of the elemental sensitivity numbers and the element change between no neighboring materials. Additionally, the addition ratio’s value reduction strategy assures the convergence of the thermal expansion properties to the desired value. The homogenization method is used to obtain the equivalent thermal expansion properties of the designed materials. Some numerical examples are presented to show the potential and effectiveness of the proposed methodology.

Microstructures, i.e., architected materials, are designed today, typically, by maximizing an objective, such as bulk modulus, subject to a volume constraint. However, in many applications, it is often more appropriate to impose constraints on other physical quantities of interest.
In this paper, we consider such generalized microstructural optimization problems where any of the microstructural quantities, namely, bulk, shear, Poisson ratio, or volume, can serve as the objective, while the remaining can serve as constraints. In particular, we propose here a neural-network (NN) framework to solve such problems. The framework relies on the classic density formulation of microstructural optimization, but the density field is represented through the NN’s weights and biases.
The main characteristics of the proposed NN framework are: (1) it supports automatic differentiation, eliminating the need for manual sensitivity derivations, (2) smoothing filters are not required due to implicit filtering, (3) the framework can be easily extended to multiple-materials, and (4) a high-resolution microstructural topology can be recovered through a simple post-processing step. The framework is illustrated through a variety of microstructural optimization problems.

Recent advances in multifunctional material technologies have paved the way for the creation of innovative multifunctional structures. Exploring multifunctional structures with advanced functionalities is a major step toward a new era of autonomous structural systems for future smart cities. Autonomous structures can respond to their environment, self-monitor their condition, process and store information, and self-program themselves. This chapter presents the state-of-the-art technologies required to build multifunctional material and structures. The central role of structure-dominated, scale-independent architected materials in developing novel types of performance-tailored multifunctional structures is highlighted. We then present a concept toward the next stage of the technological revolution in multifunctional structure science where the so-called engineered self-aware structures can sense, empower, and program themselves using their constituent components. Experimental studies are conducted using composite beam prototypes designed under the proposed concept. We highlight the capabilities of these multifunctional structures to serve as mechanically tunable, self-powered distributed sensing networks with energy harvesting functionally for smart cities infrastructure systems.

This paper develops a multi-scale topology optimization method that realizes optimized structural stiffness design while achieves inter-connectivity among the heterogeneous unit cells. Specifically, about the technical details, lattice structure topology optimization (LSTO) is conducted by optimizing the parameter field of the specially-designed multi-variable lattices, through which the optimized lattice parameters reflect the density and stress states of the associated macro-element. Then, the macro-elements with close lattice parameters are gathered into clusters, providing the initial guess for the next-step freeform optimization. Finally, multiscale topology optimization (MTO) through the inverse homogenization approach is performed to further design the unit cell structures. The unit cell structures for each cluster are forced to be identical to save homogenization-related computational resources and the interconnectivity is ensured due to the optimized and perfectly connected initial guess from LSTO. Using the proposed method, three classical numerical examples are studied that prove the effects of improved mechanical performance, ensured micro-structure inter-connectivity, and the affordable computing scale. Finally, mechanical tests are conducted to verify the design performance benefits of the proposed method.

Multi-scale topology optimization (a.k.a. micro-structural topology optimization, MTO) consists in optimizing macro-scale and micro-scale topology simultaneously. MTO could improve structural performance of products significantly. However, a few issues related to connectivity between micro-structures and high computational cost have to be addressed, without resulting in loss of performance. In this paper, a new efficient multi-scale topology optimization (EMTO) framework has been developed for this purpose. Connectivity is addressed through adaptive transmission zones which limit loss of performance. A pre-computed database of micro-structures is used to speed up the computing. Design variables have also been chosen carefully and include the orientation of the micro-structures to enhance performance. EMTO has been successfully tested on two-dimensional compliance optimization problems. The results show significant improvements compared to mono-scale methods (compliance value lower by up to 20% on a simplistic case or 4% on a more realistic case), and also demonstrate the versatility of EMTO.

This paper presents a general formulation and solution method for the problem of two-scale concurrent topology optimization of cellular structure and its anisotropic materials. This formulation seeks simultaneous determination of the macro and micro structural topologies and the orientations of microstructures, such as a unit cell with anisotropic and inhomogeneous material properties. In the present formulation, the microstructure of cellular material is uniform in the entire macrostructure but with spatially-varying orientations. The solution method consists of two new features: (a) a mutual strain energy density-based approach proposed for analytically determining the orientations of fully anisotropic materials in compliant mechanism problems, which can also be modified to address minimum compliance designs and (b) an extended and fully-coupled moving iso-surface threshold (MIST) method and algorithm developed for solving the formulated concurrent optimization problem of cellular structures using two-scale physical response functions. The present formulation and algorithm are validated via studying numerical examples of concurrent optimum design of: (i) macro and micro topologies; (ii) macro topology and orthotropic material orientations; and (iii) macro and micro topologies and material orientations.

To realize extraordinary wave phenomena, metamaterials need to attain unique effective material properties. In this work, we propose an inverse design strategy for metamaterials with specific anisotropic EMD (effective mass density). Although the conventional inverse homogenization technique has been extended to various fields, few works have been published to explore the inverse realization of an EMD tensor, each component of which is supposed to gain a given value at a target frequency. To this end, we propose a calculation scheme, in which the EMD tensor can be calculated in a much similar way to the homogenized static stiffness. Therefore, the scheme is quite convenient for sensitivity analysis. The coating layer interfacing the core and matrix is chosen as the design region because it directly determines the motion of the core. The matrix layout, which not only contributes to the stiffness of the metamaterial but also highly affects the core's local motion, is chosen carefully. The perfect transmodal Fabry–Pérot interference phenomenon is considered in this work, and through several numerical examples, the phenomenon is ideally realized. The proposed design strategy could be critically useful in designing locally resonant metamaterials with general anisotropy.

Scale separation is often assumed in most homogenization-based topology optimization (TO) frameworks for design of material microstructures. This work goes beyond the mainstream TO contributions by abandoning the scale separation hypothesis. First, it puts to evidence the limits of the homogenization-based approach when the size of the Representative Volume Element (RVE) is not negligible with respect to the structure. Then, a re-localized scheme bridging the RVE and the structure is proposed to reproduce the microscopic fields, while the structure problem at the macroscopic scale is solved only based on the coarse mesh. Finally, numerical experiments show interesting results on 2D lattice structures within the proposed framework giving a hint towards a feasible realization of the finite-scale lattice structures with current resolution of additive manufacturing technologies. Reported results evidence that the present method can lead to the same topology and stiffness of the optimized structures as the reference solution when the number of unit cell is relatively large, while reducing the computational costs significantly.

Two of the most prominent two‐dimensional auxetic mechanisms, namely the rotating rigid unit and the chiral systems, work, at least partly, by converting linear to rotational motion. This property can be harnessed by introducing a system that can convert back the rotational motion to a linear displacement so as to induce an expansion or a compression in the out‐of‐plane direction. The push drill mechanism offers an effective way of providing this capability. It is hence possible to couple two‐dimensional rotating rigid unit or chiral structures to the push drill mechanism to create novel 3D auxetic structures. The resultant Poisson’s ratio in the out‐of‐plane direction can be both negative and positive depending on the connectivity. In this work, the concept is illustrated with the use of rotating squares. An analytical model is derived to determine the parameters on which the Poisson’s ratio and Young’s moduli depend. Subsequently, the analytical results are compared to those obtained from experimental testing of different sample structure with the two showing good agreement. Further considerations indicate that it is possible to design 2D and 3D auxetic systems with the push drill as the only deformation mechanism.
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In this paper, we proposed a new material design method by microstructure topology optimization. Novelty of the proposed method is to target the whole nonlinear volume-averaged effective stress-strain curve of microstructure representative volume element (RVE) rather than aiming specific values such as strength, stiffness or Poisson ratio. J2 plasticity model with a linear isotropic hardening model was chosen for local residuals. Global residuals are computed within nonlinear finite element framework for the topology optimization. Sensitivities of the objective function augmented with the residuals and adjoint response vectors with respect to design variables are derived with details and their numerical computational procedures were also presented. Microstructure topologies showing two different targeted stress-strain curves under uniaxial and biaxial loadings were obtained by using the method of moving asymptotes (MMA) optimization algorithm. Accuracy of the sensitivity computations was verified and numerical examples demonstrated a potential of the proposed method in applications to multiscale topology optimization.

Effective cooling or heat exchange is a typical engineering issue related with various industrial products, and the lattice structure fabricated by additive manufacturing is expected to be useful for effective liquid cooling. Moreover, such a heat exchanger demands structural performances such as stiffness and small thermal deformation when, for example, casting die and transporters of heated objects. Thus, in this research, we develop an optimization method for lattice volume fraction distribution using lattice structure approximation and a gradient method considering three coupled physical problems: fluid flow, thermal conduction, and convection and linear elasticity. Fluid flow is approximated by deriving effective properties from the Darcy–Forchheimer law and analyzing the flow according to the Brinkman–Forchheimer equation. The basis lattice shape is formed as three orthogonally connected pillars. The effective performance of a representative dimension unit lattice was calculated based on the statically averaging theorem and the relationship between the design variable and effective properties were approximated by polynomial functions. Two types of optimization problems were considered: maximization of fluid cooling performance under strain energy constraint and unconstrained minimization of normal direction of the loading and heating surface. The validity of the proposed methodology was investigated through three-dimensional examples. Although observable errors in accuracy exist between results obtained from optimization and full-scale models, the relative performance optimization was considered successful.

The development of cooling devices is important for many industrial products, and the lattice structure fabricated by additive manufacturing is expected to be useful for effective liquid cooling. However, lattice density should be carefully designed for an effective arrangement of coolant flow. In this research, we optimize the lattice density distribution using a lattice structure approximation and the gradient method. Fluid flow is approximated by deriving effective properties from the Darcy–Forchheimer law and analyzing the flow according to the Brinkman–Forchheimer equation. Thermal conduction and convection are also approximated as a weakly coupled problem. We use a simple basic lattice shape composed of pillars, optimizing only its density distribution by setting the pillar diameter as the design variable. Steady-state pressure and temperature reductions are treated as multi-objective functions. Through 2D and 3D numerical studies, we discuss the validity and limitations of the proposed method. Although observable errors in accuracy exist between the results obtained from the optimization and full scale models, relative performance optimization was considered successful.

A concurrent optimization design method for the topologies of structures and materials and the material orientation is presented based on bi-direction evolutionary structural optimization (BESO) method. The macrostructure is assumed to be composed of a uniform cellular material but with different orientation. The homogenization technique is used to calculate the effective properties of the cellular material which builds a connection between material and structure. An analytical method, which is flexible to deal with the shear “weak” and “strong” materials, is proposed to solve the material orientation optimization problem. The optimization algorithm considering the simultaneous optimization of topologies of macrostructures and material microstructures, and material orientations is developed. Numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm and show that concurrent topology design of structures and materials with material orientation optimization can greatly improve the structural performance.

In the past decade, mechanical metamaterials have garnered increasing attention owing to its novel design principles which combine the concept of hierarchical architecture with material size effects at micro/nanoscale. This strategy is demonstrated to exhibit superior mechanical performance that allows us to colonize unexplored regions in the material property space, including ultrahigh strength‐to‐density ratios, extraordinary resilience, and energy absorption capabilities with brittle constituents. In the recent years, metamaterials with unprecedented mechanical behaviors such as negative Poisson's ratio, twisting under uniaxial forces, and negative thermal expansion are also realized. This paves a new pathway for a wide variety of multifunctional applications, for example, in energy storage, biomedical, acoustics, photonics, and thermal management. Herein, the fundamental scientific theories behind this class of novel metamaterials, along with their fabrication techniques and potential engineering applications beyond mechanics are reviewed. Explored examples include the recent progresses for both mechanical and functional applications. Finally, the current challenges and future developments in this emerging field is discussed as well. Mechanical metamaterials are known to be able to exhibit unique behaviors and structural properties that breach classical theories. In this review, the core building blocks behind mechanical metamaterials are explored, along with the associated state‐of‐the art design, modeling, fabrication/manufacturing, testing and characterization techniques. Emerging engineering applications across multidisciplinary fields and future developments in various industries are also discussed.

Three 3D models with negative compressibility have been presented in our previous paper [X. Q. Zhou et al., Phys. Status Solidi B 2016, 253, 1977]. However, the 2D mechanism (i.e., wine‐rack mechanism) which we used to design 3D models is so symmetrical that the axial properties of it are equivalent, so we have only focused on geometry features. In this work, an extended study is conducted in order to find a more efficient method to design 3D structures with evident negative compressibility. Through theoretical modeling, the compressibility properties of three 3D models made from hinging hexagonal truss mechanism are analyzed and the conditions for negative compressibility to be exhibited are discussed. The study shows that in addition to the geometry features, negative compressibility effects can also be significantly affected by the arrangement of the framework and layout orientations of 2D mechanism. The compressibility properties of three 3D models made from hinging hexagonal truss mechanism are analyzed and the conditions for the negative compressibility to be exhibited are discussed. It is shown that in addition to the geometry features which have already been widely discussed, the arrangement of the framework and the layout orientations of 2D mechanisms can also have a great influence on the negative compressibility effect.

Three‐dimensional printing (3DP) is a recognized additive manufacturing (AM) or rapid prototyping technology, which allows the manufacturer to construct custom 3D objects using computer software and computer‐aided design. This chapter describes various materials used for AM and provides an array of information on the properties of materials used in 3D and four‐dimensional printing. The most promising applications of 3DP have been reported in the area of biomedical engineering including human health. Rheology measures the flow and behavior of the materials. Ceramics comprise both metallic and nonmetallic elements and have been used as materials for 3D printed scaffolds because of their high mechanical strength and biocompatibility. Polymers with the low melting point are extensively used in 3DP because of their low weight, low cost, and processing flexibility. Bioprinting technologies allow the automated biofabrication of cell‐laden constructs through the layer‐by‐layer deposition of bioinks in both in vivo and in vitro.

A structural material optimization method with varied displacement constraints is proposed to solve the problem of structural material optimization with displacement constraints based on the structural topology optimization idea. By using the conventional fractional function filter functions, the effective stiffness matrix and its derivatives are established by treating the reciprocal topological variables of micro structural elements as design variables, and the one order approximate explicit functions of displacement constraints are constructed. Integrated with the idea of varied displacement constraints, a topological optimization model of micro structures is formed by treating the structural mass as objective function and the displacement as constraint functions. Then, a dual solving method is adopted. Several typical numerical examples are presented to validate the feasibility and effectiveness of the proposed optimization algorithm and a variety of microstructures of cellular materials are obtained.

IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References

Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.

This paper aims to develop a level-set-based topology optimization approach for the design of negative permeability electromagnetic metamaterials, where the topological configuration of the base cell is represented by the zero-level contour of a higher-dimensional level-set function. Such an implicit expression enables us to create a distinct interface between the free space and conducting phase (metal). By seeking for an optimality of a Lagrangian functional in terms of the objective function and the governing wave equation, we derived an adjoint system. The normal velocity (sensitivity) of the level-set model is determined by making the Eulerian derivative of the Lagrangian functional non-positive. Both the governing and adjoint systems are solved by a powerful finite-difference time-domain algorithm. The solution to the adjoint system is separated into two parts, namely the self-adjoint part, which is linearly proportional to the solution of the governing equation; and the non-self-adjoint part, which is obtained by swapping the locations of the incident wave and the receiving planes in the simulation model. From the demonstrative examples, we found that the well-known U-shaped metamaterials might not be the best in terms of the minimal value of the imaginary part of the effective permeability. Following the present topology optimization procedure, some novel structures with desired negative permeability at the specified frequency are obtained.

Evolutionary structural optimization (ESO) method was originally developed based on the idea that by system-atically removing the inefficient material, the residual shape of the structure evolves toward an optimum. This paper presents an extension of the method called bidirectional ESO (BESO) for topology optimization subject to stiffness and displacement constraints. BESO allows for the material to be added as well as to be removed to modify the structural topology. Basic concepts of BESO including the sensitivity number and displacement extrapolation are proposed and optimization procedures are presented. Integrated with the finite element analysis technique, BESO is applied to several two-dimensional plane stress problems. Its effectiveness and efficiency are examined in comparison with the results obtained by ESO. It is found that BESO is more reliable and computationally more efficient than ESO in most cases. Its capability and limitation are discussed. Nomenclature C = mean compliance E = Young's modulus K* = element stiffness matrix / = thickness of plate Uj = displacement at the constrained location M* = limit of the displacement constraint ii' = element displacement vector due to real load u ij -element displacement vector due to unit virtual load acting at the location of the displacement constraint W = weight of current structure W^ = weight of maximum structure V^ b j = objective weight WSbT (1) = first local minimum of the objective weight \y|) T b 1 j l(2) = second local minimum of the objective weight W 0?t -weight of optimal topology WQ = weight of structure of the full design area W* = target weight a = sensitivity number A = increment v = Poisson's ratio

This paper introduces a hierarchical concurrent design approach to maximizing the natural frequency of a structure. Multiple material phases are considered in the topology optimization performed on both the macro and micro scales. A general problem for composite structure and material design is formulated that contains the cellular design problem as a special case. The design of the macro structure and material micro structure is coupled. The designed material properties are applied to the analysis of the macro structure, while the macro structure displacement field is considered in the sensitivity analysis on the micro scale. The material edistribution is controlled by an optimality criterion for frequency maximization. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are employed to obtain the final optimal solution. Several numerical examples of composite structures and materials are presented to demonstrate the capability and effectiveness of the proposed approach. Results include various orthotropic or anisotropic composite materials, as well as vibration-resisting layouts of the macro structure. In-depth discussions are also given on the effects of the base material phases and the assignment of the volume fractions on each scale. (c) 2013 Elsevier Ltd. All rights reserved.

Design of functionally graded material (FGM), in which the mechanical property varies along one direction, is the focus of this study. It is assumed that the microstructure of the FGM is composed of a series of base cells in the variation direction and self-repeated in other directions. Bi-directional evolutionary structural optimization technique in the form of inverse homogenization is used for the design of the FGM for specified variation in bulk or shear modulus. Instead of designing a series of base cells simultaneously, the base cells are optimized progressively by considering three base cells at each stage. Thus, the proper connections between adjacent base cells can be achieved with high computational efficiency. Numerical examples demonstrate the effectiveness of the proposed method for designing microstructures of 2D and 3D FGMs with specified variation in bulk or shear modulus. The proposed algorithm can also be easily extended to design FGMs with other functional properties.

An orthotropic material is characterized by nine independent moduli. The ratios between the Young’s moduli in three directions are indicative of the level of orthotropy and the bulk modulus is indicative of the overall stiffness. In this paper we propose a method for designing the stiffest orthotropic material which has prescribed ratios for Young’s moduli. The material is modeled as a microstructure in a periodic unit cell. By using the homogenization method, the elasticity tensors are calculated and its compliance matrix is derived. A Lagrangian function is constructed to combine the objective and multiple equality constraints. To enable a bi-section search algorithm, the upper and lower bounds on those multipliers are derived by using a strain energy approach. The overall optimization is based on the bi-directional evolutionary structural optimization (BESO) method. Examples of various orthotropy ratios are investigated. The topology presents a constant pattern of material re-distributed along the strongest axis while the overall stiffness is maintained.

Most materials compress axially in all directions when loaded hydrostatically. Contrary to this, some materials have been discovered that exhibit negative linear compressibility and, as such, expand along a specific axis or plane. This paper analyses a fundamental mechanism by using a combination of finite element simulations and analytical derivations to show that negative linear compressibility can be found in a body-centred or face-centred tetragonal network of nodes connected by a network of beams. The magnitude and direction of this behaviour depends on the cross geometry in the network.

This paper presents a new approach to designing periodic microstructures of cellular materials. The method is based on the bidirectional evolutionary structural optimization (BESO) technique. The optimization problem is formulated as finding a micro-structural topology with the maximum bulk or shear modulus under a prescribed volume constraint. Using the homogenization theory and finite element analysis within a periodic base cell (PBC), elemental sensitivity numbers are established for gradually removing and adding elements in PBC. Numerical examples in 2D and 3D demonstrate the effectiveness of the proposed method for achieving convergent microstructures of cellular materials with maximum bulk or shear modulus. Some interesting topological patterns have been found for guiding the cellular material design.

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.

The linear compressibility of a solid is defined as the relative decrease in length of a line when the solid is subjected to unit hydrostatic pressure. Materials with a negative linear or area compressibility could have interesting technological applications. However, in the case of homogeneous materials only rare crystal phases exhibit this effect. In particular, for isotropic or cubic solids the linear compressibility is known to be isotropic and positive, namely a sphere of a cubic or isotropic crystal under hydrostatic pressure remains a sphere. For less symmetric solids, it generally varies with the direction n. Here we derive explicit expressions of the stationary values (maximum and minimum) of linear compressibility for single phase solids with monoclinic, orthotropic, tetragonal, trigonal, and hexagonal symmetry. A list of crystals that may exhibit negative linear compressibility in certain directions is outlined. Next, by assembling a two-component material, we propose microstructure networks to achieve such a property. Numerical simulations, based on a refined finite element method, are provided.

Materials exhibiting negative linear compressibility display the very unusual and unexpected property of expanding in at least one direction when placed under compressive hydrostatic stress. Here, it is shown that this property may be manifested by systems having high positive Poisson's ratios (non-auxetic), including non re-entrant hexagonal honeycombs and wine-rack models where deformation primarily involves changes in the angles between the ribs of the structures. (C) 2011 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

A three-dimensional cellular system that may be made to exhibit some very unusual but highly useful mechanical properties, including negative Poisson's ratio (auxetic), zero Poisson's ratio, negative linear and negative area compressibility, is proposed and discussed. It is shown that such behaviour is scale-independent and may be obtained from particular conformations of this highly versatile system. This model may be used to explain the auxetic behaviour in auxetic foams and in other related cellular systems; such materials are widely known for their superior performance in various practical applications. It may also be used as a blueprint for the design and manufacture of new man-made multifunctional systems, including auxetic and negative compressibility systems, which can be made to have tailor-made mechanical properties.

The counterintuitive phenomenon of negative linear compressibility (NLC) is a highly desirable but rare property exploitable in the development of artificial muscles, actuators and next-generation pressure sensors. In all cases, material performance is directly related to the magnitude of intrinsic NLC response. Here we show the molecular framework material zinc(II) dicyanoaurate(I), Zn[Au(CN)(2)](2), exhibits the most extreme and persistent NLC behaviour yet reported: under increasing hydrostatic pressure its crystal structure expands in one direction at a rate that is an order of magnitude greater than both the typical contraction observed for common engineering materials and also the anomalous expansion in established NLC candidates. This extreme behaviour arises from the honeycomb-like structure of Zn[Au(CN)(2)](2) coupling volume reduction to uniaxial expansion, and helical Au…Au 'aurophilic' interactions accommodating abnormally large linear strains by functioning as supramolecular springs.

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.

A 3D hybrid zinc formate framework, [NH(4)][Zn(HCOO)(3)], possessing an acs topology, shows a high degree of mechanical anisotropy and negative linear compressibility (NLC) along its c axis. High-pressure single-crystal X-ray diffraction studies and density functional theory calculations indicate that contraction of the Zn-O bonds and tilting of the formate ligands with increasing pressure induce changes in structure that result in shrinkage of the a and b axes and the NLC effect along c.

We consider the discretized zero-one continuum topology optimization problem of finding the optimal distribution of two linearly
elastic materials such that compliance is minimized. The geometric complexity of the design is limited using a constraint
on the perimeter of the design. A common approach to solve these problems is to relax the zero-one constraints and model the
material properties by a power law which gives noninteger solutions very little stiffness in comparison to the amount of material
used.
We propose a material interpolation model based on a certain rational function, parameterized by a positive scalar q such
that the compliance is a convex function when q is zero and a concave function for a finite and a priori known value on q.
This increases the probability to obtain a zero-one solution of the relaxed problem.

A finite element model for three-dimensional, open-called foams is developed. A reentrant cell shape is defined that has the unusual property of exhibiting a negative Poisson's ratio. By altering the cell shape it is possible to achieve different values for the elastic constants and different degrees of anisotropy. It is shown that simple two-dimensional approximations fail to model the properties of the three-dimensional network accurately and a full three-dimensional analysis is required. The reentrant cell used is fully auxetic, with negative Poisson's ratios exhibited when loaded in all three orthogonal directions.

The vast majority of materials shrink in all directions when hydrostatically compressed; exceptions include certain metallic or polymer foam structures, which may exhibit negative linear compressibility (NLC) (that is, they expand in one or more directions under hydrostatic compression). Materials that exhibit this property at the molecular level--crystalline solids with intrinsic NLC--are extremely uncommon. With the use of neutron powder diffraction, we have discovered and characterized both NLC and extremely anisotropic thermal expansion, including negative thermal expansion (NTE) along the NLC axis, in a simple molecular crystal (the deuterated 1:1 compound of methanol and water). Apically linked rhombuses, which are formed by the bridging of hydroxyl-water chains with methyl groups, extend along the axis of NLC/NTE and lead to the observed behavior.

A simple material can shrink in one direction when heated, and expand in that direction when squeezed by hydrostatic pressure.

Silver(I) hexacyanocobaltate(III), Ag3[Co(CN)6], shows a large negative linear compressibility (NLC, linear expansion under hydrostatic pressure) at ambient temperature at all pressures up to our experimental limit of 7.65(2) GPa. This behavior is qualitatively unaffected by a transition at 0.19 GPa to a new phase Ag3[Co(CN)6]-II, whose structure is reported here. The high-pressure phase also shows anisotropic thermal expansion with large uniaxial negative thermal expansion (NTE, expansion on cooling). In both phases, the NLC/NTE effect arises as the rapid compression/contraction of layers of silver atoms—weakly bound via argentophilic interactions—is translated via flexing of the covalent network lattice into an expansion along a perpendicular direction. It is proposed that framework materials that contract along a specific direction on heating while expanding macroscopically will, in general, also expand along the same direction under hydrostatic pressure while contracting macroscopically.
• negative linear compression
• negative thermal expansion
• high-pressure
• framework materials
• flexibility

Rare crystal phases that expand in one or more dimensions when hydrostatically compressed are identified and shown to have
negative Poisson's ratios. Some of these crystals (i) decrease volume and expand in two dimensions when stretched in a particular
direction and (ii) increase surface area when hydrostatically compressed. Possible mechanisms for achieving such negative
linear and area compressibilities are described for single crystals and composites, and sensor applications are proposed.
Materials with these properties may be used to fabricate porous solids that either expand in all directions when hydrostatically
compressed with a penetrating fluid or behave as if they are incompressible.