Evaluation of the in-plane effective elastic moduli of single-layered graphene sheet

In this paper, an equivalent continuum-structural mechanics approach is used to characterize the mechanical behaviour of nanostructured graphene. The in-plane elastic deformation of armchair graphene sheets is simulated by using finite element modelling. The model is based on the assumption that force interaction among carbon atoms can be modelled by load-carrying beams in a representative two-dimensional honeycomb lattice structure. The elastic properties of beam elements are determined by equating the energies of the molecular structure and the continuum beam model subjected to small strain deformation. Then an equivalent continuum technique is adopted to estimate effective elastic moduli from which elastic constants are extracted. A comparison of elastic constants obtained from current modelling concur with results reported in literature. With the multifunctional properties of graphene sheets as manifested in a broad range of industrial applications, determination of their elastic moduli will facilitate a better design of the corresponding materials at macroscopic level.     

Chirality-dependent anisotropic elastic properties of a monolayer graphene nanosheet

An analytical approach is presented to predict the elastic properties of a monolayer graphene nanosheet based on interatomic potential energy and continuum mechanics. The elastic extension and torsional springs are utilized to simulate the stretching and angle variation of carbon-carbon bond, respectively. The constitutive equation of the graphene nanosheet is derived by using the strain energy density, and the analytical formulations for nonzero elastic constants are obtained. The in-plane elastic properties of the monolayer graphene nanosheet are proved to be anisotropic. In addition, Young's moduli, Poisson's ratios and shear modulus of the monolayer graphene nanosheet are calculated according to the force constants derived from Morse potential and AMBER force field, respectively, and they were proved to be chirality-dependent. The comparison with experimental results shows a very agreement.     

The limits of Poisson's ratio in polycrystalline bodies

While it has been established that the elastic moduli and compliances of anisotropic and isotropic materials should be positive for thermodynamic reasons, no condition related to the values of Poisson's ratio has yet been established. However, it is generally accepted that for isotropic materials Poisson's ratio should vary between — 1.0 and 0.5, whereas for orthotropic materials various conditions have been introduced relating the different components of the anisotropic Poisson's ratio with the remaining elastic constants of the material. In this paper, limits for Poisson's ratio of body-centred cubic (bcc) polycrystalline materials are determined, based on the modes of deformation of a typical unit cell of the material subjected to a uniform external loading arbitrarily oriented relative to the principal axes of the crystal. It is shown that the values of Poisson's ratio thus established correlate satisfactorily with experimental values of this constant. The procedure can be readily applied to other structural units of polycrystalline bodies.     

The in-plane linear elastic constants and out-of-plane bending of 3-coordinated ligament and cylinder-ligament honeycombs

Four novel cylinder-ligament honeycombs are described, where each cylinder has 3 tangentially-attached ligaments to form either a hexagonal or re-entrant hexagonal cellular network. The re-entrant cylinder-ligament honeycombs are reported for the first time. The in-plane linear elastic constants and out-of-plane bending response of these honeycombs are predicted using finite element (FE) modelling and comparison made with hexagonal and re-entrant hexagonal honeycombs without cylinders. A laser-crafted re-entrant cylinder-ligament honeycomb is manufactured and characterized to verify the FE model. The re-entrant honeycombs display negative Poisson’s ratios and synclastic curvature upon out-of-plane bending. The hexagonal and ‘trichiral’ honeycombs possess positive Poisson’s ratios and anticlastic curvature. The ‘anti-trichiral’ honeycomb (short ligament limit) displays negative Poisson’s ratios when loaded in the plane of the honeycomb, but positive Poisson’s ratio behaviour (anticlastic curvature) under out-of-plane bending. These responses are understood qualitatively through considering deformation occurs via direct ligament flexure and cylinder rotation-induced ligament flexure.     

On the Relationship Between Microstructure of the Cortical Bone and its Overall Elastic Properties

The relationship between microstructure of the cortical bone and its effective elastic properties is discussed. We utilize results of Kachanov et al (1994) on materials with cracks/pores of diverse shapes. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are compared with available experimental data. For Young's moduli and shear moduli the agreement is good, whereas Poisson's ratios differ significantly. Possible reasons for this difference are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Microstructural modelling of auxetic microporous polymers

A simple two-dimensional model for the deformation of auxetic microporous polymers (those with a negative Poisson's ratio) has been developed. This model network of rectangular nodules interconnected by fibrils has been further developed to include the possibilities of fibril hinging, flexure and stretching. Expressions for strain-dependent Poisson's ratio and Young's modulus have been derived and compared with experimental results on microporous PTFE and UHMWPE. A combination of the hinging mode followed by the stretching mode of deformation can be used to explain the general features of the experimental data for these auxetic polymers. The force coefficients governing the different modes of deformation are dependent on fibril dimensions and intrinsic material properties. By varying the geometry of the network, the model can be used to predict different combinations of Poisson's ratio with modulus, from large positive through to large negative values.     

蜂窝纸板静态弹性与黏弹性特性建模与参数识别

目的研究蜂窝纸板的静态力学行为。方法对蜂窝纸板试样进行压缩-恢复实验,用多项式表达试样的弹性力,用分数阶微分模拟试样的黏弹性力。考虑到试样变形和弹性力的对称性,根据压缩和恢复阶段的黏弹性力之差,利用信赖域方法识别试样的黏弹性参数,再进行试样的弹性特性参数识别。结果实验与模拟结果对比表明,提出的模型可准确模拟蜂窝纸板的力学行为,最大响应误差不超过7%。结论可利用三次多项式和五参数粘弹性模型分析蜂窝纸板的静态特性,为分析静态条件下蜂窝纸板的力学特性,研究蜂窝纸板的应力和应变的时变特性,优化包装设计方法提供了参考。     

The elastic properties, elastic models and elastic perspectives of metallic glasses

Bulk metallic glass (BMG) provides plentiful precise knowledge of fundamental parameters of elastic moduli, which offer a benchmark reference point for understanding and applications of the glassy materials. This paper comprehensively reviews the current state of the art of the study of elastic properties, the establishments of correlations between elastic moduli and properties/features, and the elastic models and elastic perspectives of metallic glasses. The goal is to show the key roles of elastic moduli in study, formation, and understanding of metallic glasses, and to present a comprehensive elastic perspectives on the major fundamental issues from processing to structure to properties in the rapidly moving field.A plentiful of data and results involving in acoustic velocities, elastic constants and their response to aging, relaxation, applied press, pressure and temperature of the metallic glasses have been compiled. The thermodynamic and kinetic parameters, stability, mechanical and physical properties of various available metallic glasses especially BMGs have also been collected. A survey based on the plentiful experimental data reveals that the linear elastic constants have striking systematic correlations with the microstructural features, glass transition temperature, melting temperature, relaxation behavior, boson peak, strength, hardness, plastic yielding of the glass, and even rheological properties of the glass forming liquids. The elastic constants of BMGs also show a correlation with a weighted average of the elastic constants of the constituent elements. We show that the elastic moduli correlations can assist in selecting alloying components with suitable elastic moduli for controlling the elastic properties and glass-forming ability of the metallic glasses, and thus the results would enable the design, control and tuning of the formation and properties of metallic glasses.We demonstrate that the glass transition, the primary and secondary relaxations, plastic deformation and yield can be attributed to the free volume increase induced flow, and the flow can be modeled as the activated hopping between the inherent states in the potential energy landscape. We then propose an extended elastic model to understand flow in metallic glass and glass-forming supercooled liquid, and the model presents a simple and quantitative mathematic expression for flow activation energy of various glasses. The elastic perspectives, which consider all metallic glasses exhibit universal behavior based on a small number of readily measurable parameters of elastic moduli, are presented for understanding the nature and diverse properties of the metallic glasses.     

Analysis of equivalent elastic modulus of asymmetrical honeycomb

In this report, elastic moduli of honeycomb consisting of asymmetrical hexagonal cells are studied by using a theoretical approach and the finite element method (FEM). Based on the change in the shape of the hexagonal cell, explicit equations describing the equivalent elastic moduli of honeycomb with cell wall parallel to the y-axis are proposed. In the analysis of honeycomb deformation, the shear deformation was considered in addition to bending deformation and tensile deformation. As a result, the equivalent elastic moduli could be calculated with extremely high precision.     

自相似多级纳米蜂窝铝结构力学性能的分子动力学模拟

首次采用分子动力学方法预测了自相似多级纳米蜂窝铝面内和面外（轴向）的压缩力学性能（弹性模量和压缩强度）。重点研究了相对密度、层级数和长度比对自相似多级纳米蜂窝铝结构力学性能的影响。在Gibson模型中引入了表面效应因子,结果表明修正的Gibson-Ashby模型与分子动力学计算结果更加吻合。此外,通过比较一级、二级和三级纳米蜂窝铝结构的变形机制发现,二级和三级纳米蜂窝铝结构由于分别在单级蜂窝和二级蜂窝的角点处接入六边形,在压缩过程中,多级纳米蜂窝铝结构激发的位错远高于单级蜂窝铝结构。也就是说,在压缩载荷下,多级蜂窝铝结构可以更好地利用结构的承载能力,吸收更多的能量。但是,自相似纳米蜂窝铝结构的力学性能无法通过增加级数的方法来无限增强,在相对密度和长度比不变的情况下,当纳米蜂窝铝结构的级数达到二级时,其综合力学性能最佳。研究结果还表明,相对密度不变时,二级纳米蜂窝铝结构长度比分别在0.3和0.4附近时,二级蜂窝铝结构具有最佳的面内和面外力学性能。研究成果对自相似多级纳米蜂窝结构的优化设计具有重要的指导作用。      

Modelling the effects of negative Poisson's ratios in continuous-fibre composites

Finite-element methods have been used to examine the elastic properties of continuous-fibre reinforced composites. Consideration has been given to the possibilities of using either reinforcing or matrix constituents with large negative Poisson's ratios (with values of –1.0 v –0.3). It is shown that a large negative Poisson's ratio can lead directly to considerably enhanced transverse moduli without altering the longitudinal moduli. For example, changing the matrix Poisson's ratio from 0.3 to–0.9 and keeping all other constituent properties constant produces an almost four-fold increase in the composite transverse modulus.     

Holographic study of conventional and negative Poisson's ratio metallic foams: elasticity,yield and micro-deformation

An experimental study by holographic interferometry is reported of the following material properties of conventional and negative Poisson's ratio copper foams: Young's moduli, Poisson's ratios, yield strengths and characteristic lengths associated with inhomogeneous deformation. The Young's modulus and yield strength of the conventional copper foam were comparable to those predicted by microstructural modelling on the basis of cellular rib bending. The re-entrant copper foam exhibited a negative Poisson's ratio, as indicated by the elliptical contour fringes on the specimen surface in the bending tests. Inhomogeneous, non-affine deformation was observed holographically in both foam materials.     

The effect of honeycomb relative density on its effective in-plane elastic moduli: An experimental study

Honeycombs are discrete materials at the macro-scale level but their mechanical properties need to be calculated as a continuum material in order to simplify their design in engineering applications. The effective mechanical behavior of hexagonal honeycombs was studied by analytical means and correlated with experimental results for aluminum honeycombs. In particular, the effective in-plane elastic moduli of the honeycombs were studied as a function of their relative density. The effect of the bending, shear and axial deformations on various existing beam models was analyzed for both in-plane honeycomb directions. An experimental program was performed with honeycombs of densities ranging from the commercial low-density ones to the solid construction material. It is shown experimentally that the beam models describe well the material response in the direction of the honeycomb double wall. However, it is concluded that the effective elastic moduli for honeycombs with low relative densities are not similar in the two in-plane directions as predicted by previous studies.     

Some aspects of inverse problem analysis for a layered halfspace

The present paper discusses the questions associated with the nondestructive testing of flexible pavements. The problem of how to determine layer moduli from the measured surface deformation is solved using technique based on the theory of ill-posed problems. The proposed iterative solution process exploits the capacity to compute effectively the derivatives of the error functional with respect to elastic moduli. The properties of the procedure are analysed and tested by numerically solved example problems. It is shown that by simulated computer experiments based on the solution procedure properties, valuable information on the measuring instrumentation setup may be obtained.     

Functional grading in hierarchical honeycombs: Density specific elastic performance

The introduction of hierarchy into structures has been credited with improving their elastic and other properties. Similarly, functional grading has been demonstrated to increase the damage tolerance of honeycomb structures, although with the penalty of reduced Young’s modulus or increased density. The combination of both hierarchy and functional grading has not been reported for honeycomb structures, although it is known in natural materials. A parametric numerical modelling study has been made of the in-plane elastic properties of honeycombs and how they are affected by functional grading and hierarchy, and importantly to establish whether it is possible to avoid reductions in Young’s modulus. A set of analytical models has been developed to describe functional grading and hierarchy in honeycombs, based upon beam mechanics and the transform section method. The conditions for transition of a hierarchical honeycomb in behaviour from that of a discrete structure to that of a continuum are established. Furthermore, conditions are established for which hierarchical honeycombs, uniform or functionally graded, can surpass in-plane Young’s moduli of conventional honeycombs a by factor of up to 2, on an equal density basis.     

Room temperature single crystal elastic constants of boron carbide

Single crystals of the high temperature ceramic boron carbide have been synthesized by the optical floating zone method. Room temperature elastic constants of a carbon-deficient boron carbide single crystal have been measured using the resonant ultrasound spectroscopy technique. Based upon density measurements, the single crystal stoichiometry was specified as B_5.6C. This crystal has room temperature single crystal elastic constants of c ₁₁ = 542.81, c ₃₃ = 534.54, c ₁₃ = 63.51, c ₁₂ = 130.59, and c ₄₄ = 164.79 GPa, respectively. Analysis of Cauchy's relationships, Poisson's ratios, and elastic anisotropic factors for the single crystal elastic constants indicates that it is more strongly anisotropic in elasticity and interatomic bonding than most solids. Room temperature isotropic elastic moduli of boron carbide show that its bulk, shear and Young's moduli are substantially higher than those of most solids, so that boron carbide belongs to the so-called strong solids. Its Poisson's ratio is significantly lower than that of most solids.     

具有球形胞体结构的泡沫塑料弹性常数的确定

通过微分法导出了泡沫塑料剪切模量和体积模量所满足的微分方程组，并利用泡沫塑料各向同性弹性常数间满足的关系求解；得到了泡沫塑料剪切模量与体积模量的关系，确定了剪切模量与材料孔隙比的关系；并且将本文结果同其他已有模型了对比。     

Bending deformation of honeycomb consisting of regular hexagonal cells

In this study, the flexural rigidity of a honeycomb consisting of regular hexagonal cells is investigated. It is found that the bending deformation of the honeycomb cannot be evaluated by using the equivalent elastic moduli obtained from the in-plane deformation since the moments acting on inclined walls of honeycomb cell are different for the in-plane deformation and bending deformation. Based on the fact that the inclined wall of the honeycomb is twisted under the condition that the rotation angle in both connection edges is zero in bending deformation, a theoretical technique for calculating the honeycomb flexural rigidity is proposed. In the theoretical analysis, a torsion problem of a thin plate was solved by using the generalized variational principle. The validity of the present analysis is demonstrated by numerical results obtained by the finite element method.     

A study of the interaction between a propagating crack and an uncoated/coated elastic inclusion using the BE technique

The purpose of the present work is a parametrical study of the interaction between a propagating edge-crack and an embedded elastic fibre using the Boundary Element (BE) technique. Uniaxial fibre reinforced composites generally have very good properties in the direction of the fibres, but in conventional multi-layer crossply laminates it is cracking in the transverse direction which effectively limits the strength of a stressed body. Therefore, in this study the propagation of a crack in the transverse direction is considered, i.e., in a plane containing the fibre axes, rather than perpendicular to the fibres. Crack deflection/attraction mechanisms and their associated energy release rate variations are investigated for a range of Young's moduli and Poisson's ratio mismatches, and crack offsets with respect to the inclusion centreline. Furthermore, the effects of a third-phase, i.e., coating, applied to the fibre's surface are analysed, and results have been obtained for different coating thicknesses and elastic moduli ratios. From this investigation it was found that the Poisson's ratio of the different phases could have a significant effect on the crack trajectory, and hence the energetics involved in the process of crack deflection are also dramatically altered. This opens up the possibility of enhancing the fracture toughness of fibre reinforced composite materials by considering the Poisson's ratio of the individual phases when selecting the particular material combination.     

Elastic properties of di-ammonium hydrogen citrate single crystals: an ultrasonic study

Many organic crystals with orthorhombic symmetry are non-centrosymmetric, exhibiting electro-optic, ferroelectric and triboluminescence properties. The ammonium salt of citric acid, di-ammonium hydrogen citrate, belongs to this group, which is reported to be piezoelectric and triboluminescent. All the nine second-order elastic stiffness constants of this single crystal have been determined from ultrasonic velocity measurements employing pulse echo overlap technique. The anisotropy of elastic wave propagation in this crystal has been demonstrated by plotting the phase velocity, Young's modulus and linear compressibility surfaces. The relevant values of Poisson's ratios have been evaluated and reported. Variation of selected elastic constants with temperature have been measured and reported.