超弹性薄膜与可压缩基底双层结构表面失稳分析

当上层超弹性硬质薄膜和下层可膨胀基底构成的双层结构受压时,薄膜的自由表面可通过形成褶皱降低系统能量.研究表明,上下两层的模量比不同时,上层弹性硬质薄膜将表现出不同的表面失稳模式.本文提出了一种新颖的方法可有效抑制双层软材料的表面失稳,即改变基底材料的泊松比,这种方法同时适用于不具有应变硬化的软材料.首先基于Neo-Hookean模型发展了小变形条件下双层结构表面失稳的理论模型,通过半解析的方法得到了表面失稳的临界应变;然后通过有限元计算与模拟,进一步验证了负泊松比基底可延缓表面失稳.结果表明:(1)当双层结构基底泊松比为正且趋于0.5(不可压缩)时,双层结构在较小的压缩应变下出现表面失稳;(2)当基底的泊松比为负且趋于-1时,可被压缩至46%而不出现表面失稳,即可膨胀基底能有效抑制薄膜的表面失稳.本文发展的方法及主要结果可为延展性电子器件的设计提供指导.     

Measurement of curved-surface deformation in cylindrical coordinates

Stereo vision is used to measure the strain field of a round tension test specimen in a cylindrical coordinate system. Initially, the displacement fields of the specimen are measured relative to a world coordinate system erected by the stereo vision. Through coordinate transformations, the measured displacement fields expressed in world coordinates are then converted to the displacement fields expressed in cylindrical coordinates. By differentiating the axial and circumferential displacements in the axial and circumferential directions, the axial, circumferential and shear strains are determined. Results indicate that the measured mean value of the axial strains is in good agreement with the measurements of the extensometer and the strain gage. The Poisson's ratio obtained by the circumferential and axial strains is close to .33 in the elastic state. The mean error of the computed shear strain is approximately .03 percent in the smaller elastic deformation and .08 percent in the larger plastic deformation.     

Problems in determining the elastic strain energy function for rubber

Lightly crosslinked natural rubber can be stretched by 600% or more, and recovers almost completely. It is often regarded as a model highly elastic material and characterized by a strain energy function to describe its stress-strain behavior under various types of deformation. A number of such functions have been proposed; some of them appear in current finite element programs. They are usually validated by comparison with measured stress-strain relations by Treloar [7] [L.R.G. Treloar, Stress-strain data for vulcanized rubber under various types of deformation, Trans. Faraday Soc. 40 (1944) 59-70] and Jones and Treloar [15] [D.F. Jones, L.R.G. Treloar, The properties of rubber in pure homogeneous strain, J. Phys. D Appl. Phys. 8 (1975) 1285-1304]. But Treloar pointed out that the relations at high strains became markedly irreversible, and he did not assign a strain energy function for strains greater than about 300%. Rivlin's universal relation between torsional stiffness and tensile stress [14] [R.S. Rivlin, Large elastic deformations of isotropic materials. Part V1: further results in the theory of torsion, shear and flexure, Philos. Trans. R. Soc. A 243 (1949) 251-288] is applied here to show that a typical elastic solid cannot be described by any strain energy function at strains greater than about 300%. Elastic strain energy functions for higher strains, or for other rubbery materials, are thus of doubtful value unless evidence for reversibility of stress-strain relations is adduced or the applicability of a strain energy function is demonstrated.     

The influence of dewetting in filled elastomers on the changes of their mechanical properties

Summary The changes of mechanical properties of filled polymers and their dependence on deformation history are the subject of this paper. For most of filled polymers in practical use, the theory of linear viscoelasticity cannot be applied, even at small deformations. In this work, samples of glass bead filled polybutadiene rubber with different filler levels were investigated at small strains (<10%). The evolution of the relaxation modulus and Poisson's ratio was observed in cyclic experiments, which were also applied in inducing a defined deformation history for the succeeding relaxation experiments. In these experiments, the relaxation modulus and Poisson's ratio were measured as functions of time, with strain, strain rate, filler level and the preceeding deformation history as parameters. The results indicate dewetting as the main reason for the changes of the mechanical properties of the filled materials.The financial support of this work by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged.     

Connexions between the moduli for anistropic elastic materials

Summary For homogeneous isotropic elastic materials there are simple interrelations connecting Young's modulus, Poisson's ratio, the rigidity modulus and the modulus of compression. However for anisotropic materials the situation is quite different. Young's modulus is a function of direction and Poisson's ratio and the rigidity modulus are functions of pairs of orthogonal directions. Here some simple universal connexions between the moduli for various directions are simply derived for general anisotropic materials. No particular symmetry is assumed in the material.     

负泊松比结构力学设计、抗冲击性能及在车辆工程应用与展望

轻量化多功能负泊松比结构由于具有优异的可设计性、拉胀特性、剪切模量、断裂韧性、抗冲击吸能、减震降噪等特性,在车辆吸能结构设计和多功能优化方面具有巨大的应用潜力.本文详细综述了负泊松比结构的力学设计及其在车辆工程中的典型应用:(1)负泊松比基本概念及其力学特性,以及近几十年来的快速发展趋势;(2)负泊松比材料与结构构型设...     

用能量法分析层状材料弹性接触问题

利用能量法分析了层状材料（薄膜／基体）弹性接触问题，得到了具有一阶精度的闭合解，给出了求解薄膜弹性模量和泊松比的表达式，并与有限元的数值解进行了比较。二者比较结果表明：在工程材料范围内，理论解与数值解相差在6％以内；同时表明单相材料中剪切模量与弹性模量之间的关系也适用层状材料中的薄膜材料。在数值解的基础上，讨论了薄膜厚度与压头半径的比值对求解精度的影响，发现此比值对精度影响不大。通过对层状材料等效泊松比与等效弹性模量的定义，给出了用压痕实验测定薄膜泊松比与弹性模量的方法。     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Coupled hydro-mechanical effects in a poro-hyperelastic material

Fluid-saturated materials are encountered in several areas of engineering and biological applications. Geologic media saturated with water, oil and gas and biological materials such as bone saturated with synovial fluid, soft tissues containing blood and plasma and synthetic materials impregnated with energy absorbing fluids are some examples. In many instances such materials can be examined quite successfully by appeal to classical theories of poroelasticity where the skeletal deformations can be modelled as linear elastic. In the case of soft biological tissues and even highly compressible organic geological materials, the porous skeleton can experience large strains and, unlike rubberlike materials, the fluid plays an important role in maintaining the large strain capability of the material. In some instances, the removal of the fluid can render the geological or biological material void of any hyperelastic effects. While the fluid component can be present at various scales and forms, a useful first approximation would be to treat the material as hyperelastic where the fabric can experience large strains consistent with a hyperelastic material and an independent scalar pressure describes the pore fluid response. The flow of fluid within the porous skeleton is defined by Darcy's law for an isotropic material, which is formulated in terms of the relative velocity between the pore fluid and the porous skeleton. It is assumed that the form of Darcy's law remains unchanged during the large strain behaviour. This approach basically extends Biot's theory of classical poroelasticity to include finite deformations. The developments are used to examine the poro-hyperelastic behaviour of certain one-dimensional problems.     

基于反求的医疗辅助弹性护具应变能的计算

提出了基于反求工程的医疗辅助弹性护具的设计方法,推导出弹性护具应变能的计算公 式. 设计中应用网格简化技术减少扫描数据以及参数化技术将三维转化为二维进行计算,减 少了网格点应变能的计算量,提高了设计效率. 应用应变能密度函数建立起弹性护具几何尺 寸与应变能之间的关系,为医生或患者科学地选择医疗护具提供了依据.     

Atomistic simulation of the structure and elastic properties of gold nanowires

We performed atomistic simulations to study the effect of free surfaces on the structure and elastic properties of gold nanowires aligned in the 〈100〉 and 〈111〉 crystallographic directions. Computationally, we formed a nanowire by assembling gold atoms into a long wire with free sides by putting them in their bulk fcc lattice positions. We then performed a static relaxation on the assemblage. The tensile surface stresses on the sides of the wire cause the wire to contract along the length with respect to the original fcc lattice, and we characterize this deformation in terms of an equilibrium strain versus the cross-sectional area. While the surface stress causes wires of both orientations and all sizes to increasingly contract with decreasing cross-sectional area, when the cross-sectional area of a 〈100〉 nanowire is less than , the wire undergoes a phase transformation from fcc to bct, and the equilibrium strain increases by an order of magnitude. We then applied a uniform uniaxial strain incrementally to 1.2% to the relaxed nanowires in a molecular statics framework. From the simulation results we computed the effective axial Young's modulus and Poisson's ratios of the nanowire as a function of cross-sectional area. We used two approaches to compute the effective elastic moduli, one based on a definition in terms of the strain derivative of the total energy and another in terms of the virial stress often used in atomistic simulations. Both give quantitatively similar results, showing an increase in Young's modulus with a decrease of cross-sectional area in the nanowires that do not undergo a phase transformation. Those that undergo a phase transformation experience an increase of about a factor of three of Young's modulus. The Poisson's ratio of the 〈100〉 wires that do not undergo a phase transformation show little change with the cross-sectional area. Those wires that undergo a phase transformation experience an increase of about 10% in Poisson's ratio. The 〈111〉 wires show, with a decrease of cross-sectional area, an increase in one of Poisson's ratios and small change in the other.     

Pointwise and Other Decay Estimates for an Isotropic Elastic Strip

This paper is concerned with an homogeneous isotropic linear elastic strip, in plane strain. It is supposed that its lateral boundaries are displacement-free and that the deformation is generated by actions on the ends. A cross-sectional measure of deformation, complementing that of a previous paper, is defined and shown to satisfy a generalised convexity condition in the axial variable x ₁, for materials with negative Poisson's ratio σ. An enhanced measure is subsequently defined, and, in the case of a semi-infinite strip, is shown to yield pointwise exponential decay estimates for both the axial and the transverse displacement components for materials with positive Poisson's ratio. This revised version was published online in July 2006 with corrections to the Cover Date.     

Improved Calibration Coefficients for the Hole-Drilling Method Considering the Influence of the Poisson Ratio

For residual stress analyses with the incremental hole-drilling method adequate evaluation formalisms in order to transform the measured strains into stresses are required. The Integral Method is the most important theory used for analyses of nonlinear residual stress depth distributions. It is based on a calibration, which is carried out by Finite-Element-Analyses. For the sake of simplicity the used numerical models often represent an ideally cylindrical hole and ideal elastic material behavior with fixed elastic constants. The adaption of the calibration coefficients from the numerical simulations to the real experimental state is often performed by simple correction factors or is ignored completely. The following investigation highlights the influence of the Poisson ratio on the calibration coefficients for the most commonly used strain gage rosettes geometries, which are standardized by ASTME837-08. It comes out that the application of existing simple correction factors is only valid within a small range and better approximations can be obtained by alternative strategies.     

Elastic constant measurement by using line-focus-beam acoustic microscope

The purpose of this study is to develop a method for measuring elastic constants using a LFB acoustic microscope. For this purpose, a theoretical procedure for the estimation of Young's modulus and Poisson's ratio is introduced based on elastic theory. According to this procedure, experimental velocity measurement and elastic constants estimation are made by use of a LFB acoustic microscope. For the confirmation of the availability of this method, the estimated elastic constant is compared to measurements of elastic constants by other methods. The resulting estimated values of Young's modulus and Poisson's ratio obtained by the LFB acoustic microscope are highly accurate which confirms the usefulness of the elastic constant measurement system. T. Mihara is Research Associate, and M. Obata (former SEM Member), deceased, was Professor, Department of Materials Processing, Faculty of Engineering, Tohoku University, Aramaki Aoba, Sendai, 980, Japan.     

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.     

横观各向同性介质中椭圆夹杂受非弹性剪切变形引起的弹性场的确定

本文求解了横观各向同性介质中椭圆夹杂内受非弹性剪切变形引起的弹性场。采用各向异性弹性力学平面问题的复变函数解法,结合保角变换,获得夹杂内应变能和基体内边界的应力分布和相应的应变能的表达式。进一步,根据最小应变能原理,获得表征夹杂平衡边界的两个特征剪切应变,从而得到了弹性场的解析解。通过应力转换关系,验证了应力解满足夹杂边界上法向正应力和剪应力的连续条件,表明了该解的正确性。本文解可用于复合材料断裂强度的分析中。     

Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation

For illustrative purposes this paper treats a special problem in the theory of finite deformations of elastic materials whose associated displacement equations of equilibrium do not remain elliptic at all strains. The typical deformation arising in this problem possesses a discontinuous gradient, so that quasi-static motions involving such equilibrium states may be dissipative. For a special class of such “non-elliptic” elastic materials, it is shown that the macroscopic response in the problem treated may be precisely of the form associated with elastic—perfectly plastic behavior. The counterparts of yield, plastic strain and plastic strain rate are determined by the underlying elastic strain energy function.     

On a representation of an elastic potential in A. A. Il’yushin’s generalized strain space

For elastic isotropic materials under finite strains, we consider an elastic potential in the form of a function of invariants of the Hencky logarithmic strain measure. For such a potential, we propose a representation in A. A. Il’yushin’s generalized strain space. This representation is used to construct an approximation to the elastic potential for incompressiblematerials; this approximation permits exactly describing the stress-strain, compression, and pure shear diagrams.