Gradient Elasticity Theories in Statics and Dynamics - A Unification of Approaches

Partly in response to a communication recently published in this journal on apparent inconsistencies between certain continuum and atomistic formulations of gradient elasticity (Yang and Guo 2005), we further elaborate on this issue in view of results and works not known or not cited in the aforementioned communication. In particular, we unify the concepts and motivations of two different formats of gradient elasticity. The first format was motivated for use in statics and aims at removing strain singularities. The second format was motivated for use in dynamics and aims at describing wave dispersion. We suggest here an alternative format of gradient elasticity that is dispersive, while its static version is identical to the first format mentioned above. Also, procedures are outlined by which the higher-order coefficients can be related to micro-structural properties. Finally, solution methods are described for static and dynamic analysis.     

摩擦力和样品厚度对压痕法测量生物试样弹性的影响

生物试样的弹性测量可为生物体疾病的早期诊断和治疗提供依据。利用压痕法对生物试样的弹性进行了测量,并用有限元软件对压痕过程进行了模拟。研究发现,试样厚度对弹性测量存在影响,试样厚度越大,测量结果越接近试样真实的杨氏模量。当试样厚度为压痕深度的75倍时,测量误差仅为0.74%。又研究了压头速度对弹性测量结果的影响。研究发现,当压头速度较大时,由于摩擦力的作用,测量结果与试样弹性的真实值之间存在一定的差异。在模拟过程中添加摩擦力可准确反演试样的弹性,误差在5%以下。     

Finite elements for elasticity with microstructure and gradient elasticity

We present a general finite element discretization of Mindlin's elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three‐dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single‐element test and the patch test. Numerical results for the benchmark problem of one‐dimensional deformation show good convergence to the closed‐form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright © 2007 John Wiley & Sons, Ltd.     

随机局部弹性与分批连续进货的随机存储模型

利用随机局部弹性的概念及运算法则,研究了分批连续进货并允许缺货的存储模型中,总费用对随机最高存储量与随机采购周期的局部弹性,给出了总费用弹性的联合概率密度的一般表达式,通过实例证明了当最高存储量与采购周期的分布特性已知时,总费用的弹性分布和弹性变化范围及弹性在该变化范围的可信度。     

熔体的粘性和弹性对LDPE／PS共混物形态的影响

研究了粘性和弹性对低密度聚乙烯／聚苯乙烯共混物形态的影响。结果表明，不相容聚合物在均匀剪切流动中的分散程度以及分散相的形状与组分的粘度比、相对弹性和体积分数有关。当分散相的粘性和弹性较基相大得多时，随分散相的体积分数增加，球状的液滴形成葡萄串状。若两组分的粘性和弹性相当，在适中的混合比下，分散相产生高度变形。不管组分的粘反比和弹性比大小，若分散相的体积分数非常低，共混物的主要形态皆为分散相的球状液滴分散在基体中。     

多商品市场供需均衡的稳定性分析

应用需求弹性矩阵与供应弹性矩阵的概念,将单商品市场供需均衡的稳定性分析推广到了多商品市场的情形。     

从弹性理论探视广东中小企业发展契机

旨在为经营者对商品供求及价格分析及为企业制定发展策略提供参考.通过对弹性定义模型的分析,从价格、收入及交叉弹性入手,从市场需求及供给两个方面考虑短期和长期的因素.认为以出口为主的大多数广东中小企业应该把握经济衰退对以基础产品为主营的企业影响会较小,但国外同类商品的需求竞争会加剧这一契机,在2010年底前国家投资4万亿扩大内需的大环境下,争取从技术创新、产品结构调整等方面有所突破.     

From atomistics to continuum: Effects of a free surface and determination of surface elasticity properties

We present an analysis of surface elasticity from the Born–Oppenheimer approximation for monatomic crystals. The analysis shows that the relaxations of crystal planes parallel to a free surface can be sufficiently determined by a low-rank algebraic Riccati equation instead of a full-scale molecular dynamic (MD) simulation, and gives new restrictions on physically reasonable atomistic models and simple criteria for surface reconstructions. In the case of surface relaxations, we calculate surface elasticity properties, i.e., surface tension, residual surface stress and surface elastic stiffness tensor, from atomistic models which are compared with experimental data and prior simulation results. The formulation also proves that surface relaxations always lower surface tension and surface elastic stiffness tensor. Together with the proposed algorithm, the formulation may be useful for investigating a variety of size-dependent phenomena of nano-structures.     

Non-Singular Stresses in Gradient Elasticity at Bi-Material Interface with Transverse Crack

Bi-material interfaces are studied with cracks that end perpendicular to the interface. As is well-known, singularities in the stresses appear when classical elasticity is used. Moreover, the nature of the singularity depends on the difference in elastic constants of the two materials. In this paper, the gradient elasticity theory of Aifantis is used to remove these singularities. This is demonstrated for a range of ratios between the two Young’s moduli.     

脆性材料弹性常数的复合测量法及程序实现

指出了直接测量脆性材料弹性常数带来的困难,提出了复合测量法,详述了数据处理过程,为测量弹性常数找到了一种更为精确的新方法,在此基础上编制了C程序。     

平板颤振导数的参数弹性研究

提出了颤振导数的“参数弹性”概念;基于复模态理论,计算了平板颤振导数的平板宽度、质量、质量惯矩、竖弯频率、扭转频率和空气密度6种参数弹性;绘制了8个颤振导数的参数弹性曲线和诺模图,分析了各自特点,获得了无量纲参数对颤振导数参数弹性的影响规律。分析结果表明,新提出的参数弹性具有概念无量纲、简洁、直观、实用特点,为颤振导数参数分析提供了一种新的有效途径。     

氮化层弹性模量及残留内应力的测定

为了计算钢铁材料氮化层的弹性模量,提出了组合樑物理模型,采用等强樑的试样,测定氮化层弹性模量及残留内应力.试验结果表明:氮化层的弹性模量比母材高出约29.2%.X射线应力常数比母材提高44.4%.     

介质弹性对多重乳液分散度的影响

多重乳液的单分散性是与粒径相关的一些重要性质如乳液稳定性、乳化剂界面分配等研究工作的基础.在保证连续相与分散相的粘度比∈(0.1,1)条件下,分别以聚丙烯酰胺及甘油为连续相,以不同内相体积w1/o一级乳液为分散相,在相应流变性质研究的基础上,研究了介质的弹性对多重乳液分散度的影响,制备了分散度20%左右的拟单分散多重乳液体系.研究表明,连续相的弹性是制备拟单分散多重乳液的必要条件,分散相的弹性有利于单分散多重乳液的制备.完善了前人的相关研究结果.     

Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen's Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional (3D) asymptotic theory is reformulated for the static analysis of simply-supported, isotropic and orthotropic single-layered nanoplates and graphene sheets (GSs), in which Eringen's nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these. The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional (2D) nonlocal plate problems, the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory (CST), although with different nonhomogeneous terms. Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions, we can obtain the Navier solutions of the leading-order problem, and the higher-order modifications can then be determined in a hierarchic and consistent manner. Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.     

Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: Experiments,theory and computations

This paper examines the effective elasticity properties of a unidirectionally reinforced carbon fibre–polyester composite. A computational simulation of an experimentally determined fibre arrangement is used to derive the effective elasticity properties of the transversely isotropic composite. The computational estimates for the elastic constants are compared with several theoretical estimates for the effective elasticity properties that are based on regular arrangement of the reinforcing fibres, their volume fraction and the elasticity properties of the constituents.     

Multiparametric Biomechanical and Biochemical Phenotypic Profiling of Single Cancer Cells Using an Elasticity Microcytometer

Deep phenotyping of single cancer cells is of critical importance in the era of precision medicine to advance understanding of relationships between gene mutation and cell phenotype and to elucidate the biological nature of tumor heterogeneity. Existing microfluidic single‐cell phenotyping tools, however, are limited to phenotypic measurements of 1–2 selected morphological and physiological features of single cells. Herein a microfluidic elasticity microcytometer is reported for multiparametric biomechanical and biochemical phenotypic profiling of free‐floating, live single cancer cells for quantitative, simultaneous characterizations of cell size, cell deformability/stiffness, and surface receptors. The elasticity microcytometer is implemented for measurements and comparisons of four human cell lines with distinct metastatic potentials and derived from different human tissues. An analytical model is developed from first principles for the first time to convert cell deformation and adhesion information of single cancer cells encapsulated inside the elasticity microcytometer to cell deformability/stiffness and surface protein expression. Together, the elasticity microcytometer holds great promise for comprehensive molecular, cellular, and biomechanical phenotypic profiling of live cancer cells at the single cell level, critical for studying intratumor cellular and molecular heterogeneity using low‐abundance, clinically relevant human cancer cells.     

金属材料弹性模量的简易测量方法

根据试样的弹性变形和载荷之间的关系,使用通用量具有测量试样的有关尺寸和变有量,并以其计算和测定金属的弹性模量。     

Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams

A class of higher-order continuum theories, such as modified couple stress, nonlocal elasticity, micropolar elasticity (Cosserat theory) and strain gradient elasticity has been recently employed to the mechanical modeling of micro- and nano-sized structures. In this article, however, we address stability problem of micro-sized beam based on the strain gradient elasticity and couple stress theories, firstly. Analytical solution of stability problem for axially loaded nano-sized beams based on strain gradient elasticity and modified couple stress theories are presented. Bernoulli–Euler beam theory is used for modeling. By using the variational principle, the governing equations for buckling and related boundary conditions are obtained in conjunctions with the strain gradient elasticity. Both end simply supported and cantilever boundary conditions are considered. The size effect on the critical buckling load is investigated.     

正交异性光弹性应力分析方法研究

本文建立了正交异性光弹性应力分析的实验边界元混合解法并编制了相应的程序系统BE-MSC.该方法仅需模型边界结点处等差线一项实验数据即可实施应力的分离。它还十分适用于存在初应力的模型材料光弹性分析问题。最后对正交异性对径受压圆盘进行了光弹性应力实例分析。      

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example was finally provided and showed the efficiency of the approach. Copyright © 2011 John Wiley & Sons, Ltd.