A long-standing question is why Poisson's ratio v nearly always exceeds 0.2 for isotropic materials, whereas classical elasticity predicts v to be between -1 to 1/2. We show that the roots of quadratic relations from classical elasticity divide v into three possible ranges: -1 < v <= 0, 0 <= v <= 1/5, and 1/5 <= v < 1/2. Since elastic properties are unique there can be only one valid set of roots, which must be 1/5 <= v < 1/2 for consistency with the behavior of real materials. Materials with Poisson's ratio outside of this range are rare, and tend to be either very hard (e.g., diamond, beryllium) or porous (e.g., auxetic foams); such substances have more complex behavior than can be described by classical elasticity. Thus, classical elasticity is inapplicable whenever v < 1/5, and the use of the equations from classical elasticity for such materials is inappropriate.
We show that under tension, a classical many-body system with only isotropic pair interactions in a crystalline state can, counterintutively, have a negative Poisson's ratio, or auxetic behavior. We derive the conditions under which the triangular lattice in two dimensions and lattices with cubic symmetry in three dimensions exhibit a negative Poisson's ratio. In the former case, the simple Lennard-Jones potential can give rise to auxetic behavior. In the latter case, negative Poisson's ratio can be exhibited even when the material is constrained to be elastically isotropic.
Cork is a natural amorphous material with near-zero Poisson's ratio that is ubiquitously used for sealing glass bottles. It is an anisotropic, transversally isotropic, composite that can hardly be scaled down. Here, we propose a new class of isotropic and reusable cork-like metamaterial that is designed from an hybrid truss-lattice material to show an isotropic Poisson's ratio close to zero. Optimization is conducted using a multi-objective genetic algorithm, assisted by an elliptical basis function neural network, and coupled with finite element simulations. The optimal micro-structured metam…
The lower bound usually cited for Poisson's ratio ν is -1, derived from the relationship between ν and the bulk and shear moduli. From consideration of the longitudinal and biaxial moduli, we recently determined that the lower bound on ν for isotropic materials is actually 1/5, a value also consistent with experimental measurements on real materials. Herein we generalize this result, first by analyzing expressions for ν in terms of six common elastic constants, and then by considering arbitrary strains. The results corroborate the prior finding that 1/5 <= ν for linear elasticity to be applica…
We investigate three-dimensional left-handed and related meta-materials based on a fully symmetric multi-gap single-ring SRR design and crossing continuous wires. We demonstrate isotropic transmission properties of a SRR-only meta-material and the corresponding left-handed material which possesses a negative effective index of refraction due to simultaneously negative effective permeability and permittivity. Minor deviations from complete isotropy are due to the finite thickness of the meta-material.
Thermodynamics of crystalline materials is analyzed using strain volumes, an incremental tensorial state variable which is the volume per unit mass multiplied by the incremental strain.
The isotropic elastic moduli closest to a given anisotropic elasticity tensor are defined using three definitions of elastic distance, the standard Frobenius (Euclidean) norm, the Riemannian distance for tensors, and the log-Euclidean norm. The closest moduli are unique for the Riemannian and the log-Euclidean norms, independent of whether the difference in stiffness or compliance is considered. Explicit expressions for the closest bulk and shear moduli are presented for cubic materials, and an algorithm is described for finding them for materials with arbitrary anisotropy. The method is illus…
A possibility to realize isotropic artificial backward-wave materials is theoretically analyzed. An improved mixing rule for the effective permittivity of a composite material consisting of two sets of resonant dielectric spheres in a homogeneous background is presented. The equations are validated using the Mie theory and numerical simulations. The effect of a statistical distribution of sphere sizes on the increasing of losses in the operating frequency band is discussed and some examples are shown.
The in-plane infinitesimal deformations of graphene are well understood: they can be computed by solving the equilibrium problem for a sheet of isotropic elastic material with suitable stretching stiffness and Poisson coefficient $ν^{(m)}$. Here, we pose the following question: does the Poisson coefficient $ν^{(m)}$ affect the response to bending of graphene? Despite what happens if one adopts classical structural models, it does not. In this letter we show that a new material property, conceptually and quantitatively different from $ν^{(m)}$, has to be introduced. We term this new parameter b…
Expressions are given for the maximum and minimum values of Poisson's ratio $ν$ for materials with cubic symmetry. Values less than -1 occur if and only if the maximum shear modulus is associated with the cube axis and is at least 25 times the value of the minimum shear modulus. Large values of $|ν|$ occur in directions at which the Young's modulus is approximately equal to one half of its 111 value. Such directions, by their nature, are very close to 111. Application to data for cubic crystals indicates that certain Indium Thallium alloys simultaneously exhibit Poisson's ratio less than -1 an…
Negative thermal expansion materials are important and desirable in science and engineering applications. However, natural materials with isotropic negative thermal expansion are rare and usually unsatisfied in performance. Here, we propose a novel method to achieve negative thermal expansion via a metamaterial approach. The metamaterial is constructed with unit cells that combine bi-material strips and anti-chiral structures. Both experimental and simulation results display isotropic negative thermal expansion properties. The coefficient of negative thermal expansion of the metamaterials is d…
Layered materials span a very broad range of solids ranging from van der Waals materials to highly complex crystal structures such as clays. They are commonly believed to have highly anisotropic properties, which is essentially attributed to weak interlayer interactions. The layered Mg3Sb2 structure is currently being intensely scrutinized due to its outstanding thermoelectric properties. Based on quantitative chemical bonding analysis we unravel that Mg3Sb2 exhibits a nearly isotropic three-dimensional (3D) bonding network with the interlayer and intralayer bonds being surprisingly similar, a…
The mechanical response of isotropic elastoplastic materials containing random distributions of initially spherical voids is investigated computationally based on Fast Fourier Transform simulations. Numerical limit-analysis simulations at constant stress triaxiality allow to determine the yield surfaces, leading in particular to the determination of a Representative Volume Element size for the onset of coalescence / inhomogeneous yielding. Moreover, two different coalescence regimes are observed that differ by the presence of shearing. The yield surfaces are found to be consistent with the com…
A material exhibiting a negative Poisson's ratio is always one of the leading topics in materials science, which is due to the potential applications in those special areas such as defence and medicine. In this letter, we demonstrate a new material, few-layer orthorhombic arsenic, also possesses the negative Poisson's ratio. For monolayer arsenic, the negative Poisson's ratio is predicted to be around -0.09, originated from the hinge-like structure within the single layer of arsenic. When the layer increases, the negative Poisson's ratio becomes more negative and finally approaches the limit a…
As a basic mechanical parameter, Poisson's ratio (ν) measures the mechanical responses of solids against external loads. In rare cases, materials have a negative Poisson's ratio (NPR), and present an interesting auxetic effect. That is, when a material is stretched in one direction, it will expand in the perpendicular direction. To design modern nanoscale electromechanical devices with special functions, two dimensional (2D) auxetic materials are highly desirable. In this work, based on first principles calculations, we rediscover the previously proposed δ-phosphorene (δ-P) nanosheets [Jie Gua…
Dilational materials are stable three-dimensional isotropic auxetics with an ultimate Poisson's ratio of -1. We design, evaluate, fabricate, and characterize crystalline metamaterials approaching this ideal. To reveal all modes, we calculate the phonon band structures. On this basis, using cubic symmetry, we can unambiguously retrieve all different non-zero elements of the rank-4 effective metamaterial elasticity tensor, from which all effective elastic metamaterial properties follow. While the elastic properties and the phase velocity remain anisotropic, the effective Poisson's ratio indeed b…
Materials with a negative Poisson's ratio, also known as auxetic materials, exhibit unusual and counterintuitive mechanical behavior - becoming fatter in cross-section when stretched. Such behavior is mostly attributed to some special reentrant or hinged geometric structures regardless the chemical composition and electronic structure of a material. Here, using first principles calculations, we report a new class of auxetic single-layer two-dimensional (2D) materials, i.e., the 1T-type monolayer crystals of groups 6-7 transition-metal dichalcogenides, MX$_2$ (M = Mo, W, Tc, Re; X = S, Se, Te).…
The sensitivity of crack growth resistance to the choice of isotropic or kinematic hardening is investigated. Monotonic mode I crack advance under small scale yielding conditions is modelled via a cohesive zone formulation endowed with a traction-separation law. R-curves are computed for materials that exhibit linear or power law hardening. Kinematic hardening leads to an enhanced crack growth resistance relative to isotropic hardening. Moreover, kinematic hardening requires greater crack extension to achieve the steady state. These differences are traced to the non-proportional loading of mat…
Black phosphorene and its analogs have attracted intensive attention due to their unique puckered structures, anisotropic characteristics, and negative Poisson's ratio. The van der Waals heterostructures assembly by stacking different materials may show novel physical properties which the parent materials don't possess. In this work, the first-principles calculations were performed to study the mechanical properties of the BP/SnSe van der Waals heterostructure. Interestingly, a near-zero Poisson's ratio vzx was found in BP/SnSe heterostructure. In addition, compared to the parent materials BP …
The Poisson's ratio is a fundamental mechanical property that relates the resulting lateral strain to applied axial strain. While this value can theoretically be negative, it is positive for nearly all materials, though negative values have been observed in so-called auxetic structures. However, nearly all auxetic materials are bulk materials whose microstructure has been specifically engineered to generate a negative Poisson's ratio. In the present work, we report using first principles calculations the existence of a negative Poisson's ratio in a single-layer, two-dimensional material, black…
Conditions for a maximum or minimum of Poisson's ratio of anisotropic elastic materials are derived. For a uniaxial stress in the 1-direction and Poisson's ratio $ν$ defined by the contraction in the 2-direction, the following three quantities vanish at a stationary value: $s_{14}$, $[2νs_{15} + s_{25}]$ and $[(2 ν-1)s_{16} + s_{26}]$, where $s_{IJ}$ are the components of the compliance tensor. Analogous conditions for stationary values of Young's modulus and the shear modulus are obtained, along with second derivatives of the three engineering moduli at the stationary values. The stationary c…
This contribution investigates the extension of the microplane formulation to the description of transversely isotropic materials such as shale rock, foams, unidirectional composites, and ceramics. Two possible approaches are considered: 1) the spectral decomposition of the stiffness tensor to define the microplane constitutive laws in terms of energetically orthogonal eigenstrains and eigenstresses; and 2) the definition of orientation-dependent microplane elastic moduli. It is shown that the first approach provides a rigorous way to tackle anisotropy within the microplane framework whereas t…
The Poisson's ratio of a material characterizes its response to uniaxial strain. Materials normally possess a positive Poisson's ratio - they contract laterally when stretched, and expand laterally when compressed. A negative Poisson's ratio is theoretically permissible but has not, with few exceptions of man-made bulk structures, been experimentally observed in any natural materials. Here, we show that the negative Poisson's ratio exists in the low-dimensional natural material black phosphorus, and that our experimental observations are consistent with first principles simulations. Through ap…
An indentation experiment involves five variables: indenter shape, material behavior of the substrate, contact size, applied load and indentation depth. Only three variable are known afterwards, namely, indenter shape, plus load and depth as function of time. As the contact size is not measured and the determination of the material properties is the very aim of the test; two equations are needed to obtain a mathematically solvable system. For elastic materials, the contact size can always be eliminated once and for all in favor of the depth; a single relation between load, depth and material p…
The Poisson's ratio characterizes the resultant strain in the lateral direction for a material under longitudinal deformation. Though negative Poisson's ratios (NPR) are theoretically possible within continuum elasticity, they are most frequently observed in engineered materials and structures, as they are not intrinsic to many materials. In this work, we report NPR in single-layer graphene ribbons, which results from the compressive edge stress induced warping of the edges. The effect is robust, as the NPR is observed for graphene ribbons with widths smaller than about 10 nm, and for tensile …
In continuum mechanics, the non-centrosymmetric micropolar theory is usually used to capture the chirality inherent in materials. However when reduced to a two dimensional (2D) isotropic problem, the resulting model becomes non-chiral. Therefore, influence of the chiral effect cannot be properly characterized by existing theories for 2D chiral solids. To circumvent this difficulty, based on reinterpretation of isotropic tensors in a 2D case, we propose a continuum theory to model the chiral effect for 2D isotropic chiral solids. A single material parameter related to chirality is introduced to…
Recently, a new type of two-dimensional layered material, i.e. C3N, has been fabricated by polymerization of 2,3-diaminophenazine and used to fabricate a field-effect transistor device with an on/off current ratio reaching 5.5E10 (Adv. Mater. 2017, 1605625). Here we have performed a comprehensive first-principles study mechanical and electronic properties of C3N and related derivatives. Ab inito molecular dynamics simulation shows that C3N monolayer can withstand high temperature up to 2000K. Besides high stability, C3N is predicted to be a superior stiff material with high Young's modulus (10…
The fracture toughness for a mode I/II crack propagating in a ductile material has been subject to numerous investigations. However, the influence of the material hardening law has received very limited attention, with isotropic hardening being the default choice if cyclic loads are absent. The present work extends the existing studies of monotonic mode I/II steady-state crack propagation with the goal to compare the predictions from an isotropic hardening model with that of a kinematic hardening model. The work is conducted through a purpose-built steady-state framework that directly delivers…
Accurate simulations of isotropic permanent magnets require to take the magnetization process into account and consider the anisotropic, nonlinear, and hysteretic material behaviour near the saturation configuration. An efficient method for the solution of the magnetostatic Maxwell equations including the description of isotropic permanent magnets is presented. The algorithm can easily be implemented on top of existing finite element methods and does not require a full characterization of the hysteresis of the magnetic material. Strayfield measurements of an isotropic permanent magnet and simu…
We use molecular simulation to construct equilibrium phase diagrams for two recently introduced model materials with isotropic, soft-repulsive pair interactions designed to favor diamond and simple cubic lattice ground states, respectively, over a wide range of densities [Jain et al., Soft Matter 9 14 (2013)]. We employ free energy based Monte Carlo simulation techniques to precisely trace the inter-crystal and fluid-crystal coexistence curves. We find that both model materials display rich polymorphic phase behavior featuring stable crystals corresponding to the target ground-state structures…
To unravel how the microstructure affects the fracture surface roughness in heterogeneous brittle solids like rocks or ceramics, we characterized the roughness statistics of post-mortem fracture surfaces in home-made materials of adjustable microstructure length-scale and porosity, obtained by sintering monodisperse polystyrene beads. Beyond the characteristic size of disorder, the roughness profiles are found to exhibit self-affine scaling features evolving with porosity. Starting from a null value and increasing the porosity, we quantitatively modify the self-affine scaling properties from a…
This paper presents an auxetic medium, consisting of a two-dimensional perforated sheet where the holes are arranged in a repetitive pattern. The hexagonal disposition of the perforations makes the medium isotropic in the plane. It is shown that negative values of the Poisson's ratio can be achieved for specific values of the dimensions and orientations of the holes. The results of the numerical simulations are confirmed by experimental tests, in which the Poisson's ratio of each specimen examined is evaluated from the displacement field obtained from the Digital Image Correlation (DIC) techni…
School of Civil Engineering and Architecture, Wuhan University of Technology（武汉理工大学土木工程与建筑学院）；Sanya Science and Education Innovation Park of Wuhan University of Technology（武汉理工大学三亚科教创新园）
The local application of mechanical stress in piezoelectric materials gives rise to boundaries across which the electric polarization changes. Polarization charges appear along such polar discontinuities and the ensuing electric fields drive a charge reconstruction with the accumulation of free carriers at the boundaries. This is particularly relevant for two-dimensional materials that can sustain very large strains and display record piezoelectric responses. Here we show by first-principles simulations the emergence of one-dimensional wires of free electrons and holes along strain interfaces,…
Using bases of fourth rank tensorial bases of complete Voigt's symmetry elaborated by Walpole we obtained expressions for inverse of Young's modulus E, inverse of shear modulus G and Poisson's ratio, which depend on components of the stiffness tensor S, on direction cosines of vectors n of uniaxial load and the vector m of lateral strain with crystalline symmetry axes. Crystalline media of high and medium symmetries are considered. Such representation yields decomposition of the above elastic characteristics to isotropic and anisotropic parts. Expressions for Poisson's coefficient are well sui…
We investigate the emergence of isotropic linear elasticity in amorphous and polycrystalline solids, via extensive numerical simulations. We show that the elastic properties are correlated over a finite length scale $ξ_E$, so that the central limit theorem dictates the emergence of continuum linear isotropic elasticity on increasing the specimen size. The stiffness matrix of systems of finite size $L > ξ_E$ is obtained adding to that predicted by linear isotropic elasticity a random one of spectral norm $(L/ξ_E)^{-3/2}$, in three spatial dimensions. We further demonstrate that the elastic leng…
Magnetic isotropic NdFeB powder is processed by the following additive manufacturing methods: (i) stereolithography (SLA), (ii) fused filament fabrication (FFF), and (iii) selective laser sintering (SLS). For the first time, a stereolithography based method is used to 3D print hard magnetic materials. FFF and SLA use a polymer matrix material as binder, SLS sinters the powder directly. All methods use the same hard magnetic NdFeB powder material. Complex magnets with small feature sizes in a superior surface quality can be printed with SLA. The magnetic properties for the processed samples are…
Architected 2D lattice materials are appealing for shape-shifting applications due to the tunable sign of Poisson's ratio. It is commonly believed that the positive and negative Poisson's ratios lead to anticlastic and synclastic curvatures respectively when the material is bent in one direction. Here, taking 2D beam lattices with star-shaped unit cells as examples, we show theoretically and demonstrate experimentally that this is not always true. At a fixed Poisson's ratio, we find a transition between anticlastic and synclastic bending curvatures controlled by the beam's cross-sectional aspe…
We propose a class of auxetic three-dimensional lattice structures. The elastic microstructure can be designed in order to have omni-directional Poisson's ratio arbitrarily close to the stability limit -1. The cubic behavior of the periodic system has been fully characterized; the minumum and maximum Poisson's ratio and the associated principal directions are given as a function of the microstructural parameters. The initial microstructure is then modified into a body centered-cubic system that can achieve a Poisson's ratio lower than -1 and that can also behave as an isotropic three-dimension…
We report a three-dimensional mechanical metamaterial that simultaneously possesses negative stiffness, negative bulk modulus, and negative Poisson's ratio. This metamaterial is a periodic arrangement of binder-shell elements. Under compression, the spherical shells dent inwards which cause the material to contract in the lateral directions. At larger deformations, snap through instability occurs and the material exhibits negative incremental stiffness. Interestingly, both incremental stiffness and incremental Poisson's ratio approach negative infinity (under displacement control) when snap-ba…
