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.     

Second-Order Estimates for the Macroscopic Response and Loss of Ellipticity in Porous Rubbers at Large Deformations

This work presents the application of a recently proposed second-order homogenization method (Ponte Castañeda, 2002) to generate estimates for effective behavior and loss of ellipticity in hyperelastic porous materials with random microstructures that are subjected to finite deformations. The main concept behind the method is the introduction of an optimally selected linear thermoelastic comparison composite, which can then be used to convert available linear homogenization estimates into new estimates for the nonlinear hyperelastic composite. In this paper, explicit results are provided for the case where the matrix is taken to be isotropic and strongly elliptic. In spite of the strong ellipticity of the matrix phase, the homogenized second-order estimates for the overall behavior are found to lose ellipticity at sufficiently large compressive deformations corresponding to the possible development of shear band-type instabilities (Abeyaratne and Triantafyllidis, 1984). The reasons for this result have been linked to the evolution of the microstructure, which, under appropriate loading conditions, can induce geometric softening leading to overall loss of ellipticity. Furthermore, the second-order homogenization method has the merit that it recovers the exact evolution of the porosity under a finite-deformation history in the limit of incompressible behavior for the matrix. Mathematics Subject Classifications (2000) 49S05, 74B20, 74Q15, 74Q05.     

多孔硅橡胶有限变形的弹性行为

针对孔隙度较大 (孔隙度大于 5 0 % )的硅橡胶材料在压缩情况下的大变形 ,提出了可描述此类可压橡胶材料力学行为的应变能密度函数 ,推导了硅橡胶材料的本构方程。利用硅橡胶材料的单轴压缩实验进行了材料参数拟合 ,讨论了多孔硅橡胶的孔隙度和体积变形对压缩性能的影响     

金属基纳米复合材料等效弹性模量的均匀化方法数值模拟

均匀化理论利用位移场双尺度渐近展开建立有限元列式，本文将其与有限元通用程序相结合，应用于金属基复合材料的弹性本构数值模拟。通过对不同尺度增强相金属基复合材料等效模量的数值模拟，考察了均匀化方法的适用情况。数值计算结果表明，对常规尺度增强相金属基复合材料，均匀化方法可以较准确地预测其等效弹性模量；对纳米增强相金属基复合材料，该方法仍可给出较好的预测，但存在某种程度的系统偏差。通过对纳米尺度增强机理的分析讨论，认为纳米增强相与基体材料问的界面效应可能有别于连续介质假设，指出可以考虑采用离散原子-连续介质耦合模型改进数值模拟结果。     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials II: Axially Symmetric Deformations

In Part I of this article, we have formulated the general structure of the equations governing small plane strain deformations which are superimposed upon a known large plane strain deformation for the perfectly elastic incompressible 'modified' Varga material, and assuming only that the initial large plane deformation is a known solution of one of three first integrals previously derived by the authors. For axially summetric deformations there are only two such first integrals, one of which applies only to the single term Varga strain-energy function, and we give here the corresponding general equations for small superimposed deformations. As an illustration, a partial analysis for the case of small deformations superimposed upon the eversion of a thick spherical shell is examined. The Varga strain-energy functions are known to apply to both natural and synthetic rubber, provided the magnitude of the deformation is restricted. Their behaviour in both simple tension and equibiaxial tension, and in comparison to experimental data, is shown graphically in Part I of this paper [1]. This revised version was published online in August 2006 with corrections to the Cover Date.     

正交各向异性材料电阻应变测量的横向效应修正

本文讨论了电阻应变片测量正交各向异性材料应变的横向效应修正。将测量各向同性测量的表观弹性模量、表观泊松比推广应用于正交各向异性材料，得出了一组与广义虎克定律形式上相同的计算式，并讨论了横向效应对应力测量精度的影响。     

Determination of Early Flow Stress for Ductile Specimens at High Strain Rates by Using a SHPB

In a dynamic experiment to obtain the high-rate stress–strain response of a ductile specimen, it takes a finite amount of time for the strain rate in the specimen to increase from zero to a desired level. The strain in the specimen accumulates during this strain-rate ramping time. If the desired strain rate is high, the specimen may yield before the desired rate is attained. In this case, the strain rates at yielding and early plastic flow are lower than the desired value, leading to inaccurate determination of the yield strength. Through experimentation and analysis, we examined the validity and accuracy of the flow stresses for ductile materials in a split Hopkinson pressure (SHPB) bar experiment. The upper strain-rate limit for determining the dynamic yield strength of ductile materials with a SHPB is identified.     

Modelling of Thin Elastic Plates with Small Piezoelectric Inclusions and Distributed Electronic Circuits. Models for Inclusions that Are Small with Respect to the Thickness of the Plate

This paper is devoted to the modelling of thin elastic plates with small, periodically distributed, piezoelectric inclusions, in view of active controlled structure design. The initial equations are those of linear elasticity coupled with the electrostatic equation. Different kinds of boundary conditions on the upper faces of inclusions are considered, corresponding to different ways of control: Dirichlet, Neumann, local or nonlocal mixed conditions. We compute effective models when the thickness a of the plate, the characteristic dimension ε of the inclusions, and ε / a tend together to zero. Other situations will be considered in two forthcoming papers. This revised version was published online in August 2006 with corrections to the Cover Date.