A class of auxetic three-dimensional lattices

We propose a class of auxetic three-dimensional lattice structures. The elastic microstructure can be designed to have an omnidirectional Poisson's ratio arbitrarily close to the stability limit of −1. The cubic behaviour of the periodic system has been fully characterized; the minimum 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-centred cubic system that can achieve Poisson's ratio lower than −1 and that can also behave as an isotropic three-dimensional auxetic structure.     

Vibration of thick auxetic plates

This short communication investigates the effect of negative Poisson's ratio on the natural frequency of thick plates of arbitrary shape. Using the Mindlin plate theory, it was generally found that as the plate's Poisson's ratio becomes more negative, the Mindlin-to-Kirchhoff natural frequency ratio increases with decreasing rate. Upon comparing (a) the use of the simplified constant shear correction factor and the more accurate variable shear correction factor, (b) with and without rotary inertia, it was found that all the four combinations stated in (a) and (b) do not give appreciable difference when the Poisson's ratio of the plate is positive. However in the case of plates with negative Poisson's ratio, results reveal that when at least one of the simplifying assumptions is used, the Mindlin-to-Kirchhoff natural frequency ratio is overestimated, and that the overestimation further increases when both the simplifying assumptions are used. When benchmarked against Reddy plate theory, the use of variable shear correction factor has almost the same effect as the inclusion of rotary inertia. Hence the use of either variable shear correction factor or rotary inertia is proposed for modeling the vibrational frequencies of conventional and auxetic isotropic plates.     

Out-of-plane modulus of semi-auxetic laminates

Materials that possess negative Poisson's ratio are termed “auxetic solids”. The out-of-plane modulus of a laminate consisting of alternating positive and negative isotropic laminas (semi-auxetic laminate) is investigated in this paper. It is herein shown that the use of the inverse rule-of-mixture for obtaining the out-of-plane Young's modulus of a laminate is valid only for conventional laminates and fully auxetic laminates. The Young's modulus by inverse rule-of mixture significantly underestimates the out-of-plane Young's modulus of a semi-auxetic laminate. It is also shown that under certain conditions, the out-of-plane Young's modulus of a semi-auxetic laminate exceeds even the direct rule-of-mixture. A correction term is developed herein for incorporation into the inverse rule-of-mixture.     

Modified virtual internal bond model based on deformable Voronoi particles

In last time, the series of virtual internal bond model was proposed for solving rock mechanics problems. In these models, the rock continuum is considered as a structure of discrete particles connected by normal and shear springs(bonds). It is well announced that the normal springs structure corresponds to a linear elastic solid with a fixed Poisson ratio, namely, 0.25 for threedimensional cases. So the shear springs used to represent the diversity of the Poisson ratio.However, the shearing force calculation is not rotationally invariant and it produce difficulties in application of these models for rock mechanics problems with sufficient displacements. In this letter, we proposed the approach to support the diversity of the Poisson ratio that based on usage of deformable Voronoi cells as set of particles. The edges of dual Delaunay tetrahedralization are considered as structure of normal springs(bonds). The movements of particle's centers lead to deformation of tetrahedrals and as result to deformation of Voronoi cells. For each bond, there are the corresponded dual face of some Voronoi cell. We can consider the normal bond as some beam and in this case, the appropriate face of Voronoi cell will be a cross section of this beam. If during deformation the Voronoi face was expand, then, according Poisson effect, the length of bond should be decrees. The above mechanism was numerically investigated and we shown that it is acceptable for simulation of elastic behavior in 0.1–0.3 interval of Poisson ratio. Unexpected surprise is that proposed approach give possibility to simulate auxetic materials with negative Poisson's ratio in interval from –0.5 to –0.1.     

星型负泊松比多孔材料力学性能及应用研究

论文推导了星型负泊松比多孔材料的泊松比$弹性模量及相对密度的解析表达式"并通过有限元分 析验证了解析表达式的准确性:星型负泊松比多孔材料具有优异的吸能与隔振性能"论文设计了一种船用星型多 孔材料隔振基座:建立了不同层数$壁厚$泊松比的星型多孔材料隔振基座对应的数值有限元模型"探究了星型多 孔材料薄壁结构泊松比$层数以及壁厚对多孔材料隔振基座强度与减振性能的影响:研究表明"减少多孔材料薄壁 结构的层数和壁厚"选用胞元泊松比?%:$的星型多孔材料"可以获得低频隔振效果好$强度高的多孔材料隔振 基。     

Scaling laws of nanoporous gold under uniaxial compression: Effects of structural disorder on the solid fraction,elastic Poisson's ratio,Young's modulus and yield strength

In this work the relationship between the structural disorder and the macroscopic mechanical behavior of nanoporous gold under uniaxial compression was investigated, using the finite element method. A recently proposed model based on a microstructure consisting of four-coordinated spherical nodes interconnected by cylindrical struts, whose node positions are randomly displaced from the lattice points of a diamond cubic lattice, was extended. This was done by including the increased density as result of the introduced structural disorder. Scaling equations for the elastic Poisson's ratio, the Young's modulus and the yield strength were determined as functions of the structural disorder and the solid fraction. The extended model was applied to identify the elastic–plastic behavior of the solid phase of nanoporous gold. It was found, that the elastic Poisson's ratio provides a robust basis for the calibration of the structural disorder. Based on this approach, a systematic study of the size effect on the yield strength was performed and the results were compared to experimental data provided in literature. An excellent agreement with recently published results for polymer infiltrated samples of nanoporous gold with varying ligament size was found.     

EFFECTS OF POISSON’S RATIO ON SCALING LAW IN HERTZIAN FRACTURE

In this paper the Auerbach＇s scaling law of Hertzian fracture induced by a spherical indenter pressing on a brittle solid is studied. In the analysis, the singular integral equation method is used to analyze the fracture behavior of the Hertzian contact problem. The results show that the Auerbach＇s constant sensitively depends on the Poisson＇s ratio, and the effective Auerbach＇s domain is also determined for a given value of the Poisson＇s ratio.     

Analytical parametric analysis of the contact problem of human buttocks and negative Poisson's ratio foam cushions

Analytical investigations on the contact problems between two homogeneous and isotropic soft bodies were performed to simulate the contact of human buttocks and seat cushions. The cushion materials' Poisson's ratio were allowed to be negative. The human buttocks were modeled as an ideal sphere with radius 15 cm, and assumed to have a low Young's modulus and a Poisson's ratio close to 0.5. These parameters were held constant during our analysis. Peak contact pressure was reduced by adjusting the contour curvature of cushions according to Hertz theory, as expected. Moreover, analysis by both the Hertz model and a finite thickness 3D elasticity model showed that using negative Poisson's ratio cushions could further reduce the pressure. Negative Poisson's ratio cushions may be beneficial in the prevention of pressure sores or ulcers in the sick and in reduction of pressure-induced discomfort in seated people.     

Linear Contact Between Plates and Unilateral Elastic Supports*

Abstract

The problem of frictionless linear contact between elastic line supports (beam or Winkler-type) and an elastic plate that is loaded laterally is considered. Contact is called linear if there is no initial gap or no initial pre-force on the potential contact surface. The shear deformation of the plate is taken into account. The contact problem is discretized at points on the potential contact surface. Numerical results are calculated with an optimization algorithm that is based on the force method. The effects of the thickness and Poisson's ratio of the plate and the elastic constants of the line supports are considered. Two loading cases are dealt with: a point load in the middle of the plate and a uniform load over the plate. The plate is rectangular, with side ratios 1 and 2.     

Measuring Poisson's ratios in wood

Poisson's ratios and Young's moduli were measured for five sugar pine (Pinus lambertiana Douglas) boards in a plane delineated by the grain direction and a tangent to the growth increments. One test specimen per board was loaded in a direction parallel to the grain; a second one was loaded perpendicular to the grain in the tangential direction. Test results from four of the five boards fulfilled theoretical expectations with a maximum deviation of ±8 percent for the quotients of Poisson's ratios divided by Young's moduli. Linear relationships were found to exist between the rate of stress parallel to the load, the rate of strain parallel to the load, and the rate of strain perpendicular to the load. With increasing magnitudes of strain rate, values for Poisson's ratios and Young's moduli approached limit values. It was speculated that at least some of the elastic constants are time dependent. However, the term in the equations which describes that property had large errors.     

Dynamic behaviour of a thin laminated plate embedded with auxetic layers subject to in-plane excitation

The dynamic modelling of a simply-supported thin laminated plate subject to in-plane excitation is established based on the classic shear theory and von Kármán nonlinear theory. The method of multiple scales is used to determine an approximate solution for the system. According to solvability conditions, the nonlinear modulation equations arising from the principal parametric resonances are obtained and two possible nontrivial solutions are performed. To analyze the nonlinear dynamic response of the plate embedded with auxetic layers, 5-layered sandwich plate, in which two auxetic elastic layers are alternatively sandwiched between three positive Poisson’s ratio (PPR) elastic ones, is presented. The natural frequency of model (m, n) shows an increase with respect to the absolute value of Poisson’s ratio. Particularly, the amplitude-frequency responses of the laminated plate subject to principal parametric resonance are analyzed for different values of Poisson’s ratio. Moreover, it can be found that for model (m, n), there must be some certain value or interval of negative Poisson’s ratio (NPR), which, results in zero response effect, in other words, the in-plane excitation will be ineffective for this model when the Poisson’s ratio just lies at such a value or interval. Furthermore, it can also be observed that the certain interval of Poisson’s ratio becomes wider with the increase of damping.     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.     

Deformation of a slab on a rigid half space subjected to a periodic strip loading

The deformation of an elastic layer bonded to a rigid half-space and subjected to bands of pressure spaced periodically along a strip is treated. It is shown that there exists a critical periodicity of the band spacing for which the displacement of the top surface, midway between the bands, is a maximum. It is concluded that this periodicity varies with the thickness of the slab, the Poisson's ratio of the material, and the band width. Numerical calculations are presented to show the effect of these parameters.     

Volume changes in hydrogels subjected to finite deformations

Constitutive equations are derived for the elastic response of hydrogels under an arbitrary deformation with finite strains. An expression is proposed for the free energy density of a hydrogel based on the Flory concept of a network of flexible chains with constrained junctions whose reference configuration differs from the initial configuration of a fully swollen gel. Adjustable parameters in the stress–strain relations are found by fitting observations on poly(acrylamide) and gellan hydrogels under uniaxial tension and compression. The effect of elongation ratio on osmotic Poisson's ratio is examined numerically.     

Role of elasticity in elastic–plastic fracture: Analytical bi-linear crack-tip fields and finite element analysis

A solution for Model-I plane strain crack tip fields in a bi-linear elastic–plastic material is presented. The elastic–plastic Poisson's ratio is introduced to characterize the influence of elastic deformation on the near tip constraint. Attention is focused on the distribution of elastic/plastic strain energy in the sensitive region of the forward sector ahead of a crack tip. The present study shows that the elastic strain energy can be higher than the plastic strain energy in this sensitive sector while large amount of the plastic strain energy develops outside this sector around the crack tip. The effect of elastic deformation in this sensitive region on the structure of crack-tip fields is considerable and the assumption in some important solutions for crack-tip fields reported in literature that the elastic deformation is small and can be ignored is therefore not physically reasonable. Besides, finite element analysis is carried out to validate the analytical solution and good agreement between them is found. It is seen that the present solution with T-stress can properly describe the crack-tip fields under various constraints for different specimens and an analytical relation is established between the critical value of J-integral, J_c, and T-stress for elastic–plastic fracture.     

Determination of young's modulus or Poisson's ratio using eddy currents

Combined a-c and d-c magnetic fields cause strains in a conducting sample. The amplitude of this strain is dependent on the value of Young's modulus and Poisson's ratio. The strain amplitude, being in the order of 10 pm, can be measured with a stabilized Michelson interferometer, described elsewhere.^1,2 An expression is derived, relating the axial strain in cylindrical samples to the magnetic-field quantities, the elastic properties and the electrical resistivity of the sample. The finite-element method is used to treat more complicated configurations. Samples of aluminum, copper, gold and tin are used for comparing the measured and calculated results. To this end, the elastic properties of the copper samples were also determined from measurement of the ultrasonic-wave velocity. The agreement between both methods is very satisfactory.     

An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage

A general closed-form solution for the so-called rebound indentation test’ is obtained for a cylindrical flat-ended punch indenting a linear viscoelastic layer lying on a rigid substrate. Under the assumption of time-independent Poisson's ratio, we derive closed-form analytical expressions for the contact force (in a displacement controlled regime) and for the indentation displacement (in a load-controlled regime) and we consider in detail the case of standard viscoelastic solid. Our results indicate that the rebound displacement (in other words the indentation displacement in the load-controlled stage) is independent of the relaxed elastic modulus and Poisson's ratio, and also of the layer's thickness. Our analytical solution can be used for layered samples of arbitrary materials exhibiting viscoelastic properties; however, since the rebound indentation test has been recently suggested for assessing the viability of biomedical materials, we have applied our theoretical framework to the identification of materials parameters from experiments on articular cartilage. In this context, we have found a pretty good agreement for the rebound deformation, even until the strain becomes relatively large.     

Effect of porosity on the properties of strain localization in porous media under undrained conditions

Within the general framework of mixture theory and by introducing the fictitious “fluid phase” as a mixture of a liquid and a gas, the conditions for localization of deformation into a shear band in the incremental response of partially saturated and fully saturated elastic–plastic porous media under undrained conditions are derived. The effect of porosity is included in the derivation. The explicit analytical expressions of the direction of shear band initiation and the corresponding hardening modulus of the porous media for the plane strain case are deduced, and a parametric analysis is made of the influence of the porosity on the properties of strain localization based on Mohr–Coulomb yield criterion. It is found that the dependence of the shear banding properties of partially saturated porous media on the porosity is related to the stress states and Poisson's ratio. However, the properties of the strain localization for the fully saturated porous media are almost independent of Poisson's ratio. Finally, on the basis of Mohr–Coulomb yield criterion, some solutions of the shear banding orientation for water-saturated granular materials are obtained, which are proved to be in good agreement with the experimental results reported by other researchers.     

增强六手臂缺失支柱手性拉胀超材料力学性能理论研究

前期研究工作中, 基于有限元分析, 作者发展了一种在大变形范围内具有可调恒定负泊松比的新型增强六手臂缺失支柱手性拉胀超材料. 为了揭示微观结构?力学性能关系, 并进一步指导超材料目标参数设计, 本文在小变形框架下基于能量法建立了表征该拉胀材料等效泊松比和弹性模量的理论模型. 增强六手臂缺失支柱手性拉胀材料由“Z”型手臂元件组成. “Z”型手臂可以被假设为两端简支的欧拉?伯努利梁. 因此, 本文首先推导了两端受集中力和力偶的任意形状欧拉?伯努利梁的应变能. 然后, 考虑平衡条件和变形协调条件进一步给出了材料等效泊松比和弹性模量的理论表达式. 研究表明只有“Z”型梁的内外手臂比为2:1时, 理论表达式才有简洁的形式. 为了更好地利用所推导的理论表达, 基于理论推导, 本文开发了MATLAT图形用户界面 (GUI). 在GUI中输入可描述该超材料几何形状的独立几何参数, 即可直接获取其等效泊松比和弹性模量. 最后, 基于理论结果, 系统讨论了超材料微结构几何参数对其等效力学性能的影响, 并将理论解与有限元计算结果进行了对比. 结果表明, 可以通过调控微结构几何参数获取大范围的目标力学性能.      

On edge dislocation in a multilayered laminate

We investigate the elastic field induced by an edge dislocation in a multilayered laminated composite composed of (N ? 2) thin bonded elastic layers sandwiched between two semi-infinite elastic media. A simple closed-form solution is obtained when all the phases have equal shear modulus but different Poisson's ratios, and when the dislocation is located in the upper semi-infinite phase. The image force acting on the dislocation due to its interaction with the multilayered structure is also derived. Several specific examples are discussed in detail to illustrate the mobility of the edge dislocation. Some interesting behaviors of the dislocation are observed. Our results indicate that it is possible to find at most (N ? 2) equilibrium positions for the edge dislocation in an N-phase composite structure.