Discrete element modeling of the effect of particle size distribution on the small strain stiffness of granular soils

Discrete element modeling was used to investigate the effect of particle size distribution on the small strain shear stiffness of granular soils and explore the fundamental mechanism controlling this small strain shear stiffness at the particle level. The results indicate that the mean particle size has a negligible effect on the small strain shear modulus. The observed increase of the shear modulus with increasing particle size is caused by a scale effect. It is suggested that the ratio of sample size to the mean particle size should be larger than 11.5 to avoid this possible scale effect. At the same confining pressure and void ratio, the small strain shear modulus decreases as the coefficient of uniformity of the soil increases. The Poisson’s ratio decreases with decreasing void ratio and increasing confining pressure instead of being constant as is commonly assumed. Microscopic analyses indicate that the small strain shear stiffness and Poisson’s ratio depend uniquely on the soil’s coordination number.     

FE analysis of the in-plane mechanical properties of a novel Voronoi-type lattice with positive and negative Poisson’s ratio configurations

This work presents a novel formulation for a Voronoi-type cellular material with in-plane anisotropic behaviour, showing global positive and negative Poisson’s ratio effects under uniaxial tensile loading. The effects of the cell geometry and relative density over the global stiffness, equivalent in-plane Poisson’s ratios and shear modulus of the Voronoi-type structure are evaluated with a parametric analysis. Empirical formulas are identified to reproduce the mechanical trends of the equivalent homogeneous orthotropic material representing the Voronoi-type structure and its geometry parameters.     

硬夹心矩形夹层板的整体稳定性分析

摘要：本文在Reissner型理论给出的位移模式基础上,修正其软夹心假设,考虑夹心层面内刚度,给出了硬夹心夹层板的几何方程、物理方程,建立了硬夹心夹层板结构在面内纵向载荷作用下的平衡微分方程,并对方程进行了简化,通过理论计算得到了四边简支条件下硬夹心矩形夹层板整体失稳临界载荷的解析解,并分别计算了夹心层材料的弹性模量 、厚度 、泊松比 对硬夹心夹层板临界载荷的影响,结果证明,对于硬夹心夹层结构,夹心层面内刚度对硬夹心夹层板整体失稳临界载荷的影响较大,考虑其面内刚度是必要的。     

Homogenization of a graphene sheet

A continuum model for a graphene sheet undergoing infinitesimal in-plane deformations is derived by applying the arguments of homogenization theory. The model turns out to coincide with that found by various authors with different methods, but it avoids anticipations on the validity of any properly adjusted or generalized Cauchy–Born rule. The constitutive equation for stress and the effective Young’s modulus and Poisson ratio are explicitly given in terms of the bond constants.     

Analytical solution of rectangular plate with in-plane variable stiffness

The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.     

多层氧化石墨烯的三维弹性常数及氧化度对力学性能的影响

宏观氧化石墨烯膜由多层石墨烯组成,其法向拉伸和层间剪切性能远比面内性能低。本文视多层氧化石墨烯为一种特殊的三维正交各向异性材料——横观各向同性材料,通过建立羟基和环氧基在石墨烯表面随机分布的多层氧化石墨烯三维模型,采用分子动力学方法模拟多层氧化石墨烯的面内拉伸、法向拉伸和层间剪切行为,分别得到了多层氧化石墨烯材料的全部五个独立弹性常数E2、E3、μ12、μ32和G23,进而确定了三维弹性矩阵（柔度矩阵和刚度矩阵）,并进一步分析了氧化度对弹性常数和强度的影响规律。结果表明：随着氧化度R逐步增大,多层氧化石墨烯面内杨氏模量E2和拉伸强度σ2max逐步降低,法向杨氏模量E3和拉伸强度σ3max、层间剪切模量G23和剪切强度τ23max均逐步增大,而对泊松比的影响较小;拉伸和剪切断裂破坏位置由氧化基团(羟基和羧基)与碳原子结合键能大小所决定。     

On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network

The numerical analysis performed here, using a finite element network model, provides a number of important results regarding the evolution of micro fractures in planar random fiber networks where the only active microscopic fracture mechanism is bond fracture. The fibers are randomly distributed in the network meaning that the network is considered having in-plane isotropic properties on the macroscopic scale. The network is loaded so that, in an average sense, homogenous macroscopic stress and strain fields are present.Several conclusions are drawn. It is found that the development of macroscopic material degradation follows an exponential two-parameter law, consisting of an onset parameter and a fracture rate parameter, justifying a previous theory derived by the authors. The fracture rate parameter is linearly related to the inverse of the bond density above a certain density limit (percolation) and increases with increasing slenderness ratio of the fibers when keeping the bond density at a constant level. The strain energies stored in interfiber bonds are exponentially distributed over the whole network. The numerical analysis reveals that there is a linear relation between the ratio of fractured and initial number of loaded bonds, and the network’s macroscopic material stiffness normalized with its pristine stiffness, confirming earlier findings based on experimental observations. At localization the analyzed theory looses its validity because the fracture process is no longer randomly distributed over the whole network. Localization coincides with location of peak load in force–displacement tensile tests.     

Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress

A modified version of the virtual internal bond model (VIB) is presented. This involves the introduction of a R-bond restricting the relative rotation freedom of pairwise mass particle. Such a modification allows the VIB model to consider arbitrary values of the Poisson ratio. A linear elastic cohesive law considering both the R-bond and L-bond are assumed. The constitutive relationship is derived using the Cauchy–Born rules. The derived constitutive associates the bond stiffness with the Young’s modulus and Poisson ratio of materials. This gives the bond stiffness in terms of the Young’s modulus and Poisson ratio of materials.The modified VIB model is then used to analyze the tensile crack behavior. In contrast to the cohesive stress method, the deformation-governed concept will be used. The local materials failure is assumed to coincide with the reduction of the bond density due to the local deformation rather than by the local cohesive stress. A phenomenological relationship between the bond density and the deformation is established. The criterion which is applied to determined crack initiation and propagation is built into the constitutive model. As an example, the method is used to study the crack initiation and propagation behavior under tensile loading.     

EFFECT OF DAMAGE EVOLUTION ON POISSON’S RATIO OF CONCRETE UNDER SULFATE ATTACK

Experimental results about concrete under sulfate attack are summarized,which include the variation of mass density of samples and velocity of ultrasonic wave propagating in samples.The evolution damage is analyzed in terms of the experimental results,and close attention is paid to the effect of damage evolution on Poisson’s ratio.This study shows that Poisson’s ratio is significantly affected by the concentration of solution and water-cement ratio.Poisson’s ratio of concrete changes very little when the water-cement ratio is selected as 0.6 or 0.8,so that such change may be neglected.If water-cement is 0.4,however,the Poisson’s ratio of the sample significantly changes.When the concrete sample of 0.4 water-cement ratio is immersed in sodium sulfate solution of 8% concentration for 285 days,Poisson’s ratio increase 10.14% compared with its initial value.There exist a sensitive region and a non-sensitive region for the change rate of Poisson’s ratio with respect to corrosion time.The change rate of Poisson’s ratio monotonously decreases with corrosion time in the sensitive region;in the non-sensitive region,the change rate of Poisson’s ratio is almost equal to zero.     

Hierarchical honeycombs with tailorable properties

We investigated the mechanical behavior of two-dimensional hierarchical honeycomb structures using analytical, numerical and experimental methods. Hierarchical honeycombs were constructed by replacing every three-edge vertex of a regular hexagonal lattice with a smaller hexagon. Repeating this process builds a fractal-appearing structure. The resulting isotropic in-plane elastic properties (effective elastic modulus and Poisson’s ratio) of this structure are controlled by the dimension ratios for different hierarchical orders. Hierarchical honeycombs of first and second order can be up to 2.0 and 3.5 times stiffer than regular honeycomb at the same mass (i.e., same overall average density). The Poisson’s ratio varies from nearly 1.0 (when planar ‘bulk’ modulus is considerably greater than Young’s modulus, so the structure acts ‘incompressible’ for most loadings) to 0.28, depending on the dimension ratios. The work provides insight into the role of structural organization and hierarchy in regulating the mechanical behavior of materials, and new opportunities for developing low-weight cellular structures with tailorable properties.     

Rayleigh and Love surface waves in isotropic media with negative Poisson’s ratio

The behavior of Rayleigh surface waves and the first mode of the Love waves in isotropic media with positive and negative Poisson’s ratio is compared. It is shown that the Rayleigh wave velocity increases with decreasing Poisson’s ratio, and it increases especially rapidly for negative Poisson’s ratios less than ?0.75. It is demonstrated that, for positive Poisson’s ratios, the vertical component of the Rayleigh wave displacements decays with depth after some initial increase, while for negative Poisson’s ratios, there is a monotone decrease. The Rayleigh waves are characterized by elliptic trajectories of the particle motion with the change of the rotation direction at critical depths and by the linear vertical polarization at these depths. It is found that the elliptic orbits are less elongated and the critical depths are greater for negative Poisson’s ratios. It is shown that the stress distribution in the Rayleighwaves varies nonmonotonically with the dimensionless depth as (positive or negative) Poisson’s ratio varies. The stresses increase strongly only as Poisson’s ratio tends to?1. It is shown that, in the case of an incompressible thin covering layer, the velocity of the first mode of the Love waves strongly increases for negative Poisson’s ratios of the half-space material. If the thickness of the incompressible layer is large, then the wave very weakly penetrates into the halfspace for any value of its Poisson’s ratio. For negative Poisson’s ratios, the Love wave in a layer and a half-space is mainly localized in the covering layer for any values of its thickness and weakly penetrates into the half-space. For the first mode of the Love waves, it was discovered that there is a strong increase in the maximum of one of the shear stresses on the interface between the covering layer and the half-space as Poisson’s ratios of both materials decrease. For the other shear stress, there is a stress jump on the interface and a more complicated dependence of the stress on Poisson’s ratio on both sides of the interface.     

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.     

Atomistic–continuum coupled model for nonlinear analysis of single layer graphene sheets

In this paper, atomistic–continuum coupled model for nonlinear flexural response of single layer graphene sheet is presented considering von-Karman geometric nonlinearity and material nonlinearity due to atomic interactions. The strain energy density function at continuum level is established by coupling the deformation at continuum level to that of at atomic level through Cauchy–Born rule. Strain and curvature dependent tangent in-plane extensional, bending–extension coupling, bending stiffness matrices are derived from strain energy density function constructed through Tersoff–Brenner potential. The finite element method is used to discretize the graphene sheet at continuum level and nonlinear bending response with and without material nonlinearity is studied. The present results are also compared with Kirchhoff plate model and significant differences at higher load are observed. The effects of other parameters like number of atoms in the graphene sheet, boundary conditions on the central/maximum deflection of graphene sheet are investigated. It is also brought out that the occurrence of bond length exceeding cutoff distance initiates at corners for CFCC, CFCF, SFSS, SFSF graphene sheets and near center for SSSS and CCCC graphene sheets.     

The theoretical model of fcc ultrathin film

A three-dimensional lattice model is constructed to theoretically study the size effects on the elastic properties of ultrathin films with face-center-cubic crystal structure. The lattice model directly takes the discrete nature in the thickness direction into account and treats the deformations along the film plane with continuum mechanics. Only the interactions between the nearest and second nearest atoms are considered in this model and represented as harmonic springs. The constitutive relation of the ultrathin film is then derived using the energy approach and the analytical expressions of the elastic moduli of ultrathin films, including in-plane, out-plane Young’s modulus and Poisson’s ratio, are obtained. Moreover, the analytical expressions of ultrathin films with different crystal orientations are also formulated. It is shown that the ultrathin film along in-plane directions may be stiffer or softer than its bulk counterpart, but it is always softer along the out-plane direction.     

Auxetic frameworks inspired by cubic crystals

In this work we show that a structure consisting of a network of bending beams can exhibit a negative Poisson’s ratio. We have shown that the negative Poisson’s ratio behaviour is driven by the (bcc analogous) type III beams, the type II (fcc like) beams result in a structure with a Poisson’s ratio of around zero and type I (simple cubic configuration) beams result in a Poisson’s ratio of nearly +1. The tensile and shear strengths of the type III beams are augmented by addition of type II and type III beams. By tailoring the relative stiffness of the component beams within the structure it is possible to design an auxetic truss structure with specific Poisson’s ratio, tensile and shear moduli.This validates the hypothesis that crystal structures can provide inspiration for macro structures with tailored mechanical properties where the mechanism for negative Poisson’s ratio (auxetic) behaviour at the atomic scale in cubic crystals is replicated by bending beams.     

The effects of hierarchy on the in-plane elastic properties of honeycombs

Introducing hierarchy into structures has been credited with improving elastic properties and damage tolerance. Specifically, adding hierarchical sub-structures to honeycombs, which themselves have good-density specific elastic and energy-absorbing properties, has been proposed in the literature. An investigation of the elastic properties and structural hierarchy in honeycombs was undertaken, exploring the effects of adding hierarchy into a range of honeycombs, with hexagonal, triangular or square geometry super and sub-structure cells, via simulation using finite elements. Key parameters describing these geometries included the relative lengths of the sub- and super-structures, the fraction of mass shared between the sub- and super-structures, the co-ordination number of the honeycomb cells, the form and extent of functional grading, and the Poisson’s ratio of the sub-structure. The introduction of a hierarchical sub-structure into a honeycomb, in most cases, has a deleterious effect upon the in-plane density specific elastic modulus, typically a reduction of 40 to 50% vs a conventional non-hierarchical version. More complex sub-structures, e.g. graded density, can recover values of density specific elastic modulus. With careful design of functionally graded unit cells it is possible to exceed, by up to 75%, the density specific modulus of conventional versions. A negative Poisson’s ratio sub-structure also engenders substantial increases to the density modulus versus conventional honeycombs.     

Compression analysis of rectangular elastic layers bonded between rigid plates

An elastic layer bonded between two rigid plates has higher compression stiffness than the elastic layer without bonding. While the finite element method can be applied to calculate the stiffness, the compression stiffness of bonded rectangular layers derived through a theoretical approach in this paper provides a convenient way for parametric study. Based on two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the compression stiffness of the bonded rectangular layer is then established in an explicit single-series form. Through the solved pressure, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear stress on the bonding surface can be found. It is found that the effect of the rectangular aspect on the compression stiffness is significant only when Poisson’s ratio is near 0.5. For the smaller Poisson’s ratio, the compression stiffness of the rectangular layer can be approximated by the formula for the infinite-strip layer of the same shape factor.     

石墨烯及其复合材料纳米力学研究进展

石墨烯是一种由碳原子构成的二维晶体,是目前已知最薄但却有着极高强度的纳米材料。由于在强度、导热性、电子输运和光学上显示出不同寻常的特性,石墨烯迅速成为材料科学、物理、化学和力学等学科的研究热点。与此同时,石墨烯复合材料的研究也迅速兴起。本文综述了近年来石墨烯及其复合材料的力学特性的研究进展。根据力学行为的差异,我们主要阐述了石墨烯面内力学特性、离面力学特性、原子尺度修饰和石墨烯复合材料力学特性的研究进展：石墨烯的面内拉伸力学特性通过纳米压痕等技术得到了测量,其断裂行为在微纳尺度下不能完全使用连续介质力学模型进行解释,在多层石墨烯情况下会出现超润滑现象;石墨烯的可控离面位移对于改变其物理特性有重要的意义,石墨烯上的屈曲受到手性和尺度的影响,在高频器件中存在着非连续性的离面响应;适当的原子尺度修饰可以改善石墨烯的拉伸和扭转力学特性;石墨烯可以改善复合材料的力学特性,如提高强度、韧性等,其主要强化效应是通过与基体材料的离面、面内力学行为结合产生的。最后,本文对石墨烯及其复合材料的力学研究进行了总结和展望。     

Stretching Graphene to 3.3% Strain Using Formvar-Reinforced Flexible Substrate

Background

As a one-atom-thick material, the mechanical loading of graphene in large scale remains a challenge, and the maximum tensile strain that can be realized is through a flexible substrate, but only with a value of 1.8% due to the weak interfacial stress transfer.

Objective

Aims to illustrate the interface reinforcement brought by formvar resins as a buffering layer between graphene and substrates.

Methods

Single crystal graphene transferred to different substrates, applied with uniaxial stretching to compare the interface strength, and finite element analysis was performed to simulate tensile process for studying the influence of Poisson’s ratio of the buffering layer for interface reinforcement.

Results

In this work we use formvar resins as a buffering layer to achieve a maximum uniaxial tensile strain of 3.3% in graphene, close to the theoretical limit (3.7%) that graphene can achieve by flexible substrate stretching. The interface reinforcement by formvar is significantly higher than that by other polymers, which is attributed to the liquid–solid phase transition of formvar for more conformal interfacial contact and its suitable Poisson’s ratio with graphene to avoid its buckling along the transverse direction.

Conclusions

We believe that these results can provide guidance for the design of substrates and interfaces for graphene loading, as well as the support for mechanics analysis of graphene-based flexible electronic devices.

     

A theoretical consideration of the ballistic response of continuous graphene membranes

The remarkable properties of graphene, including unusually high mechanical strength and stiffness, have been well-documented. In this paper, we combine an analytical solution for ballistic impact into a thin isotropic membrane, with ab initio density functional theory calculations for graphene under uniaxial tension, to predict the penetration resistance of multi-layer graphene membranes. The calculations show that continuous graphene membranes could enable ballistic barriers of extraordinary performance, enabling resistance to penetration at masses up to 100× lighter than existing state-of-the-art barrier materials. The very high elastic wave speed and strain energy to failure are the major drivers of this increase in performance. However, the in-plane mechanical isotropy of graphene, as compared to conventional orthotropic woven textiles, also contributes significantly to the efficiency of graphene as a barrier material. This result suggests that, for barrier applications, isotropic membranes composed of covalently bonded two-dimensional molecular networks could provide distinct advantages over fiber-based textiles derived from linear polymers.