STATE OF THE ART OF CONSTITUTIVE RELATIONS OF HYPERELASTIC MATERIALS 1)

超弹性材料是工程实际中的常用材料, 具有在外力作用下经历非常大变形、在外力撤去后完全恢复至初始状态的特征. 超弹性材料是典型的非线性弹性材料, 其性能可通过材料的应变能函数予以表征. 近几十年来, 围绕应变能函数形式的构造, 已提出许多超弹性材料本构关系研究的数学模型和物理模型, 但适用于多种变形模式和全变形范围的完全本构关系仍是该领域期待解决的重要问题. 本文从3个不同角度, 对超弹性材料本构关系研究的最新进展进行了总结和分析: (1)不同体积变化模式, 包含不可压与可压两种; (2)多变形模式, 包含单轴拉伸、剪切、等双轴以及复合拉剪等多个种类; (3)全范围变形程度, 包含小变形、中等变形到较大变形范围. 超弹性材料本构关系研究的最新进展表明, 为了全面描述具体材料的实验数据并在实际问题中应用超弹性材料, 需要建立适合于多种变形模式和全变形范围的可压超弹性材料的完全本构关系. 对实际超弹性材料完全本构关系的建立及可压超弹性材料应变能函数的构造, 笔者还提出了相应的实施步骤和研究方法.     

超弹性材料本构关系的最新研究进展

超弹性材料是工程实际中的常用材料, 具有在外力作用下经历非常大变形、在外力撤去后完全恢复至初始状态的特征. 超弹性材料是典型的非线性弹性材料, 其性能可通过材料的应变能函数予以表征. 近几十年来, 围绕应变能函数形式的构造, 已提出许多超弹性材料本构关系研究的数学模型和物理模型, 但适用于多种变形模式和全变形范围的完全本构关系仍是该领域期待解决的重要问题. 本文从3个不同角度, 对超弹性材料本构关系研究的最新进展进行了总结和分析: (1)不同体积变化模式, 包含不可压与可压两种; (2)多变形模式, 包含单轴拉伸、剪切、等双轴以及复合拉剪等多个种类; (3)全范围变形程度, 包含小变形、中等变形到较大变形范围. 超弹性材料本构关系研究的最新进展表明, 为了全面描述具体材料的实验数据并在实际问题中应用超弹性材料, 需要建立适合于多种变形模式和全变形范围的可压超弹性材料的完全本构关系. 对实际超弹性材料完全本构关系的建立及可压超弹性材料应变能函数的构造, 笔者还提出了相应的实施步骤和研究方法.      

可压超弹性-塑性材料中的空穴生成

本文研究了材料的弹塑性性质对球体中空穴生成问题的影响，材料的弹性用一种可压超弹性材料的本构关系来描述，材料的塑性用满足材料的不可压条件和Tresca屈服条件的理想塑性材料的本构关系来描述。这类超弹性．塑性材料中可以发生空穴的生成现象，得到了在表面拉伸作用下球体中空穴生成时空穴半径与临界拉伸之间的关系式和临界拉伸。球体的变形可分为弹-塑性变形阶段和完全塑性变形阶段，球体中心首先形成塑性变形区域，并有空穴的突然生成；塑性变形区域能够快速增长，并且使球体很快进入完全塑性变形阶段；空穴在弹-塑性变形阶段迅速增长，但进入完全塑性变形阶段后增长较慢。同时给出了不同变形阶段球体中的应力分布。数值结果表明材料的塑性性质对材料中的空穴生成有明显的影响。     

Flexible Multibody Simulation and Choice of Shape Functions

The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into a reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown coordinates depending on time only. The shape functions may be obtained using finite-element-models of flexible bodies in multibody systems, resulting in a detailed system representation and a high number of system equations. The number of system equations of such a nodal approach can be reduced considerably using a modal representation of deformation. This modal approach, however, leads to the fundamental problem of selecting the shape functions.The floating frame of reference formulation is reviewed here using a generic flexible body model, from which the various body models used in multibody simulations may be derived by formulation of specific constraint equations. Special attention is given in this investigation to the following subjects: The separation of flexible body motion into a reference motion and deformation requires the definition of a body reference frame, which in turn affects the choice of shape functions. Some alternatives will be outlined together with their advantages and disadvantages. Assuming the body deformation to be small, the system equations can be linearized. This may require considering geometric stiffening terms. The problem of how to compute these terms has been solved in literature on the instability of structures under critical loads. For finite element models the geometric stiffening terms are obtained from the tangential stiffness matrix. The generality of the flexible body model allows the definition of an object oriented data base to describe the system bodies. Such a data base includes a general interface between multibody- and finite-element-codes. By combining eigenfunctions and static deformation modes to represent body deformation one obtains a set of so-called quasi-comparison functions. When selected properly these functions can be shown to improve the representation of stresses significantly.     

The limb deformation of the compound bow

A model of the archery round-wheel compound bow with a more realistic limb deformation is introduced. The model is based on the former compound bow model of the author with the supposition that the elastic limb deformation is consistent with the Euler–Bernoulli equation. The Young’s modulus of an example bow limb is determined. The model is compared to the experimental data and to the former model which uses the modified Hickman approximation for the bow limbs. The effects of the shape of the limb on the force-draw curve of the compound bow are also considered. The model presented here may be used when designing or adjusting the limbs of the compound bow. An approximation for the path of the limb tip for the straight limb of uniform bending stiffness for small deformations is also derived.     

单向压缩状态下上海地区软土的蠕变变形与次固结特 性研究

基于上海地区原状软土的一维固结试验,讨论了在不同加荷比、加荷速率及超载预压情况下软土的蠕变变形特性和次固结特性。试验结果表明:蠕变变形随应力水平、加荷速率的增大而增大。相同应力水平下,加荷比较大的土样蠕变变形较小。蠕变变形为非线性,随着应力水平的增加,蠕变变形从粘弹性到粘塑性转变,加荷比较大时,这种转变更明显。对试验结果进行曲线拟和,确定Metchell一维经验模型参数,拟和曲线与试验结果吻合较好。通过对次固结阶段试验数据的线性拟合可知,对于正常固结土,次固结系数Ca值受应力水平和加荷比的影响不大,而与加荷速率密切相关,加荷速率较大时,Ca值随荷载的增大急剧增大后略有降低。超载预压可有效减小次固结系数,从而有利于减小工后次固结沉降。     

岩石隧道围岩变形时空效应分析

岩石隧道围岩变形具有时空效应特征。根据围岩变形速率,岩石隧道围岩变形一般可划分为3个阶段,即急剧变形阶段、稳定变形阶段和流变阶段。通过总结分析围岩变形3阶段的特点,结合中梁山隧道D-5H量测剖面的实测数据,对围岩变形的空间效应和时间效应进行了分析。空间效应集中发生在急剧变形段,空间效应段主要靠围岩自身以及初次支护克服围岩发生破坏变形,时间效应则主要体现在流变段。以华蓥山隧道等76个隧道实例为统计样本,分别对围岩变形时空效应与围岩类别和塌方事故的关系进行了相关性分析。结果表明:80％以上的塌方发生于急剧变形段,13％发生在稳定变形段,只有7％左右的塌方发生在流变段。其中Ⅳ类和Ⅴ类围岩在3个阶段都可能发生塌方,Ⅲ类围岩则很少在流变段发生塌方。Ⅰ类和Ⅱ类围岩则基本不会发生较大规模的塌方。对深入了解隧道围岩的变形规律,为隧道灾害防治、选择恰当的支护时机和支护方式很有意义。     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

An Approach to the Determination of the Deformation Characteristics of Viscoelastic Materials

An efficient method is proposed to determine the deformation function of a viscoelastic material from experimental data. The deformation function is assumed to be an integral operator with Rabotnov's fractional-exponential kernel or a sum of such kernels. This representation enables effective use of the method of operator continued fractions. To illustrate the method, deformation data for polymethylmethacrylate are used. The viscoelastic characteristics of a composite based on this material are obtained using the method of operator continued fractions __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 41–50, August 2005.     

Capturing anisotropic constitutive models with WYPiWYG hyperelasticity; and on consistency with the infinitesimal theory at all deformation levels

The elastic nonlinear behavior of fiber-reinforced materials and soft biological tissues is analyzed using anisotropic hyperelastic models. Frequently, these models are not compatible with the corresponding infinitesimal theory, but some of them may be modified to accommodate that theory in the limit. WYPiWYG hyperelasticity is compatible with the infinitesimal theory at all deformation levels and capable of capturing exactly a complete set of experimental data, which reproduces all deformation modes at every strain level, under homogeneous deformations. In this work we study the relevance of recovering the infinitesimal theory at every deformed configuration and also the performance of the WYPiWYG method in predicting the behavior of anisotropic materials at large strains under nonhomogeneous deformations.     

Constitutive relations describing creep deformation for multi-axial time-dependent stress states

A theory of primary and secondary creep deformation in metals is presented, which is based upon the concept of tensor internal state variables and the principles of continuum mechanics and thermodynamics. The theory is able to account for both multi-axial and time-dependent stress and strain states. The wellknown concepts of elastic, anelastic and plastic strains follow naturally from the theory. Homogeneous stress states are considered in detail and a simplified theory is derived by linearizing with respect to the internal state variables. It is demonstrated that the model can be developed in such a way that multi-axial constant-stress creep data can be presented as a single relationship between an equivalent stress and an equivalent strain. It is shown how the theory may be used to describe the multi-axial deformation of metals which are subjected to constant stress states. The multi-axial strain response to a general cyclic stress state is calculated. For uni-axial stress states, square-wave loading and a thermal fatigue stress cycle are analysed.     

Some remarks on the stability of the homogeneous deformation for an elastic bar

The stability of the homogeneous deformation for an elastic bar subject to uniaxial tension is studied. It is shown that the critical extension at which the homogeneous deformation becomes unstable is a decreasing function of the aspect ratio. Furthermore, it is shown that for small aspect ratios, the homogeneous deformation need not be the global minimizer of the total energy, even though it may be a local minimizer.     

新型让压锚杆作用机理研究

软岩大变形工程存在因开挖卸荷导致较大的应力释放,在工程施工过程中将会产生较大变形的特点,如沿用传统的锚杆进行支护,岩体的大变形所产生的过大拉力将会造成锚杆的拉断破坏.针对软岩大变形工程存在的该类问题,介绍了一种新型的端部具有让压装置的新型压力分散型让压锚杆,并对该让压锚杆的作用机理和力学行为进行分析,得出锚固体的作用机理与传统压力分散型锚杆相比分为3个阶段,并通过试验的方式确定了该让压锚杆的让压力.这种新型让压锚杆在将锚固体压力分散的同时,实现大变形的让压,适宜在大变形工程中进行应用.     

多孔Mooney-Rivlin材料矩形板的单向拉伸

本文利用不同可压超弹性材料大变形的Mooney-Rivlin应变能函数研究了含有多个微孔的矩形板在单向拉伸作用下的有限变形和受力分析。首先利用不可压条件得到了文中所给的含有某种对称性分布的多个微孔的矩形板的变形模式函数，其中所含的一个参数可由远离微孔的无穷远处的变形状态确定，另一个参可用最小势能原理导出变分近似解。文中详细分析了板中微孔（一个，三个和五个）随载荷作用的增长情况和微孔边缘应力的分布情况，并进行了比较。讨论了微孔的个数和排列方式，微孔的孔间距离等因素对微孔增长和应力分布的影响。     

Fibre spinning of a weakly elastic liquid

This paper presents a study of a silicone oil (poly(dimethyl siloxane)) in extensional deformation using an instrument developed recently by the authors. Data from steady shear and low amplitude sinusoidal deformation of this liquid clearly establish that it is weakly elastic. The viscometric data, for shear rates less than 100 s ⁻¹, are best represented by either the Maxwell model or the Jeffrey's model, the latter being marginally superior. The extensional data show that at low deformation rates, this fluid exhibits a Newtonian behavior with an apparent extensional viscosity equal to three times the shear viscosity. Under these conditions the velocity profiles along the spinline are also well represented by the Newtonian model. However, at higher deformation rates better predictions of the velocity profiles are obtained from the Jeffrey's and Maxwell models. At deformation rates above 100 s ⁻¹ none of these simple models is adequate. Under the conditions used in these experiments, the fractional increase in tensile stress along the fiber is shown both theoretically and experimentally to be a unique function of the total strain. Furthermore, the apparent extensional viscosity at any point on the spinline can be calculated from steady state expressions if allowance is made for the variation of stretch rates by defining a time averaged stretch rate.The results obtained here show that elasticity must be considered if these model liquids are used to conduct rheological experiments at high deformation rates. Additionally, it is found that elastic effects in extension can be predicted using simple constitutive equations provided viscometric data can be represented properly in the deformation rate range of interest. Finally, the present research further substantiates the utility of the extensional viscometer developed by the authors.     

高速冲击表面处理对金属材料力学性能和组织结构的影响

高速冲击表面处理过程中的应变率对金属材料的宏观力学性能和微观组织结构都具有重要影响。根据当前应变率效应的研究成果,从宏观与微观相结合的角度出发,综述了高速冲击表面处理过程中应变率对金属材料强度和塑性的影响规律,并重点阐述了不同应变率下金属材料内部微观组织结构的演变规律,主要包括晶粒结构、绝热剪切带、相变、位错组态和析出相以及变形孪晶等。此外,还分析了组织结构随应变率的演化和微观变形机制的转变对材料力学性能的强化和弱化机理。最后,对高速冲击表面处理梯度组织的变形特点进行了总结。提出了不同组织结构对材料性能影响的综合效应模型,以期为应变率效应的深入研究奠定基础。     

Incremental Computation Technique for Residual Stress Calculations Using the Integral Method

The Integral Method for determining residual stresses involves making surface deformation measurements within a sequence of small increments of material removal depth. Typically, the associated matrix equation for solving the residual stresses within each depth increment is ill-conditioned. The resulting error sensitivity of the residual stress evaluation makes it essential that data measurement errors are minimized and that the residual stress solution method be as stable as possible. These two issues are addressed in this paper. The proposed method involves using incremental deformation data instead of the total deformation data that are conventionally used. The technique is illustrated using an example ESPI hole-drilling measurement.     

软材料接触力学问题的实验研究

针对硫化硅橡胶在刚性压头作用下的大变形接触问题,研究了橡胶类软材料接触区区域变形场的实验测量技术。为了研究硫化硅橡胶与刚性体在试件内部接触的大变形问题,本文提出了一种体内数字栅云纹实验技术,给出了该技术的原理和实现方法。然后运用数字图像处理方法得到了橡胶材料大变形位移u场、v场,并结合橡胶材料在压头作用下的理论分区模型,分析了软材料接触区域的大变形场,讨论了接触区域的变形分区规律,并初步给出了接触区域应变计算方法。     

Fourth-order quasi-harmonic equations of state for solids of cubic and tetragonal symmetry

General expressions are given for the dependence of the pressure and the effective elastic moduli on deformation and temperature in the form of a Taylor series expansion with respect to elastic and thermal strains. The temperature dependence of these expressions is derived within the quasi-harmonic approximation of lattice dynamics. The expressions are developed in terms of the Lagrangian strain and an alternative strain measure identical with the Eulerian strain for a pure deformation. They are then used to obtain the third- and fourth-order equations of state for crystals of cubic and tetragonal symmetry and to relate the parameters entering these equations to quantities which are commonly (or may be potentially) measured experimentally. It is shown that available ultrasonic data are not completely sufficient to evaluate the parameters of fourth-order equations of state. For tetragonal symmetry, this problem is still in abeyance; while in the cubic case, it is possible to estimate the fourth-order parameters from shock-wave data and so to give illustrative numerical applications of our equations. Finally, the third- and fourth-order Hugoniots and isotherms of Cu and Ag are calculated in terms of both the Lagrangian and Eulerian strain measures.