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.     

多孔硅橡胶有限变形的粘弹性行为

针对孔隙度较大（孔隙度大于50％）的硅橡胶材料在有限变形时的粘弹性行为，从建立描述材料粘弹性特征的松驰函数和变形特征的应变能函数出发，提出了适合多孔隙、可压硅橡胶材料的非线性粘弹性力学行为的本构关系，松驰函数和应变能函数可解耦为等容和体积变形两部分，并引入了拟时间的概念来反映变形对材料特征时间的影响，利用硅橡胶材料的单轴压缩松驰实验与材料模型进行了对比，讨论了多孔硅橡胶的等容变形和体积变形对应力松驰的影响。     

非线性载荷作用下圆形薄膜的有限变形分析

利用修改后的应变能函数分析了作为细胞层载体的超弹性薄膜在非线性载荷作用下的有限变形问题.运用打靶法对控制方程进行了求解,讨论了本构参数n和α对薄膜变形的影响.计算结果表明:本构参数的增大会对材料产生强化作用;由于非线性载荷的影响,本构参数在n=1附近,薄膜曲率的变化趋势会相反;随着本构参数的减小,薄膜的不稳定点会在薄膜达到较小膨胀体积时出现.     

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     

一种描述形状记忆合金拟弹性变形行为的本构关系

本文给出一种描述形状记忆合金拟弹性变形现象的本构关系,可用于多晶材料在一般应力状态下及单晶材料在单轴应力下的变形情况。该本构关系采用弹性应变与相变应变迭加形式,物理意义明显,形式简洁。对 Cu-Zn-Sn 合金及 Ti-Ni 合金材料的变形行为进行了模拟计算,结果与实验值有较好的吻合。     

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。     

考虑缠结效应的超弹性本构模型

经典熵弹性模型, 如 Neo-Hookean模型和Arruda-Boyce八链模型, 被广泛应用于预测橡胶等软材料的超弹性力学行为. 然而, 大量实验结果也显示仅采用一套模型参数, 这类模型不能同时准确地描述橡胶在多种加载模式下的应力响应. 为了克服上述模型的不足, 本文在熵弹性的模型基础上引入缠结约束效应. 微观上, 采用Langevin统计模型来表征熵弹性变形自由能, 通过管模型(tube model)引入缠结约束自由能, 并基于仿射假设, 建立微观变形与宏观变形之间的映射关系. 在宏观上, 所建立的超弹性模型的Helmholtz自由能同时包含熵弹性和缠结约束两部分, 其中熵弹性自由能与经典的Arruda-Boyce八链模型一致, 依赖于柯西-格林应变张量的第一不变量, 而缠结约束自由能依赖于柯西-格林应变张量的第二不变量. 与文献中的实验结果对比发现, 该三参数模型能准确地预测实验中所测得的橡胶材料在单轴拉伸、纯剪切和等双轴拉伸变形条件下的应力响应, 也能较好地描述不同预拉伸比条件下双轴拉伸实验结果. 最后, 本文比较了所建立的基于应变不变量的缠结约束模型与文献中相关的缠结约束模型在多种加载模式下自由能的异同. 总的来说, 本文所建立的本构理论能准确模拟橡胶等软材料的大变形力学行为, 对其工程应用有促进作用.      

聚脲弹性体力学性能与本构关系研究进展

聚脲是一种由异氰酸酯组分和氨基组分反应生成的新型弹性体高聚物.由于聚脲具有断裂伸长率高、应变率强化、高耗能等一系列优异的力学性能,其在国防、能源、交通等领域显示出广阔的应用前景.目前,国内外学者针对聚脲在不同温度、不同应变率下的静动态力学性能开展了大量研究,在此基础上提出了多种本构模型,对温度、应变率等因素相关的力学行为进行了描述和预测.这些工作为深刻理解聚脲抗冲击机理及材料的进一步应用奠定了基础.文章首先简要介绍了聚脲弹性体的微相分离结构及特点;然后从小变形线性黏弹性和大变形非线性黏弹性两个方面概述了关于聚脲力学性能的研究,包括相应测试技术的发展和聚脲黏弹性影响因素的研究;进一步从变形梯度乘法分解法、遗传积分法、应变-时间解耦法等不同建模方法出发对已建立的聚脲本构模型进行综述,并从应变率范围、温度范围、压力相关性、软化行为表征及模型参数数量的角度对比了不同类型模型的区别;最后针对聚脲力学性能与本构关系下一步研究值得重点关注的问题提出了几点建议.     

一种描述形状记忆合金拟弹性变形行为的本构关系

本文给出一种描述形状记忆合金拟弹性变形现象的本构关系,可用于多晶材料在一般应力状态下及单晶材料在单轴应力下的变形情况。该本构关系采用弹性应变与相变应变迭加形式,物理意义明显,形式简洁。对 Cu-Zn-Sn 合金及 Ti-Ni 合金材料的变形行为进行了模拟计算,结果与实验值有较好的吻合。     

采用共旋应变的三维热弹塑性有限变形有限元法

本文采用线性化共旋应变张量和增率型虚功原理，建立了有限变形热力耦合弹塑性有限元法。在该方法中，材料的流动应力取为应变总量、应变速率和温度的函数，推导了包含这种函数关系的本构矩阵。另外在温度场分析中，考虑了塑性功和摩擦功转化的热量。文后给出的算例表明该方法可以很好地模拟热加工过程。     

一类可压缩超弹性圆柱体轴线上的空穴现象

对于由一类均匀各向同性可压缩的广义Varga材料组成的实心圆柱体,研究其在给定的外表面拉伸和轴向拉伸或压缩共同作用下的轴对称变形问题.利用能量变分原理得到了问题的控制方程和边界条件,并求得了描述柱体径向对称变形的参数型解析解和描述圆柱体轴线上空穴生成和增长的空穴分岔解.给出了与泊松比和轴向伸长相关的径向临界伸长的表达式和空穴生成后的应力表达式;并通过数值算例讨论了这些参数对圆柱体轴线上空穴生成和增长、圆柱体的径向位移以及应力的集中和突变的影响,同时给出了相应的数值模拟.     

Constitutive Models for Compressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

Constitutive models are proposed for compressible isotropic hyperelastic materials that reflect limiting chain extensibility. These are generalizations of the model proposed by Gent for incompressible materials. The goal is to understand the effects of limiting chain extensibility when the compressibility of polymeric materials is taken into account. The basic homogeneous deformation of simple tension is considered and simple closed-form relations for the deformation characteristics are obtained for slightly compressible materials. An explicit first-order approximation is obtained for the lateral contraction and for the Poisson function in terms of the axial extension which is shown to be valid for each of two specific compressible versions of the Gent model. One of the main results obtained is that the effect of limiting chain extensibility is to stiffen the material relative to the neo-Hookean compressible case. Mathematics Subject Classifications (2000) 74B20, 74G55.     

Understanding the need of the compression branch to characterize hyperelastic materials

Soft biological tissues are frequently modeled as hyperelastic materials. Hyperelastic behavior is typically ensured by the assumption of a stored energy function with a pre-determined shape. This function depends on some material parameters which are obtained through an optimization algorithm in order to fit experimental data from different tests. For example, when obtaining the material parameters of isotropic, incompressible models, only the extension part of a uniaxial test is frequently taken into consideration. In contrast, spline-based models do not require material parameters to exactly fit the experimental data, but need the compression branch of the curve. This is not a disadvantage because as we explain herein, to properly characterize hyperelastic materials, the compression branch of the uniaxial tests (or valid alternative tests) is also needed, in general. Then, unless we know beforehand the tendency of the compression branch, a material model should not be characterized only with tensile tests. For simplicity, here we address isotropic, incompressible materials which use the Valanis-Landel decomposition. However, the concepts are also applicable to compressible isotropic materials and are specially relevant to compressible and incompressible anisotropic materials, because in biomechanics, materials are frequently characterized only by tensile tests.     

Simple shearing and azimuthal shearing of an internally balanced compressible elastic material

The finite deformation response of a compressible internally balanced elastic material is studied for deformations that involve progressive shearing. The internally balanced material theory requires that an equation of internal balance is satisfied at each material point. This arises from the constitutive theory which makes use of a multiplicative decomposition of the deformation gradient. Satisfaction of the internal balance requirement then yields the most energetically favorable decomposition. Here we consider a particular compressible internally balanced material model that is motivated by a Blatz–Ko type energy from the conventional hyperelastic theory. The conventional hyperelastic theory occurs as a special limiting case of the internally balanced constitutive theory. More generally, the internally balanced material exhibits softer mechanical behavior. This gives rise to a stress-plateau in the simple shearing response whereas such plateaus do not occur in the corresponding hyperelastic treatment. The boundary value problem for azimuthal shearing with a possible radial stretching is then studied. The internally balanced material response is again found to be softer than that of the hyperelastic limiting case. This is manifest in terms of an upper bound to the applied twisting moment for the existence of solutions to the boundary value problem. In contrast, the hyperelastic limiting case has solutions for all values of applied moment.     

Distortion of Anisotropic Hyperelastic Solids Under Pure Pressure Loading: Compressibility, Incompressibility and Near-Incompressibility

The purpose of this note is to examine distortion during pure pressure loading for anisotropic hyperelastic solids. We contrast the corresponding issues in compressible and incompressible hyperelasticity, and then use these results to examine nearly incompressible materials. An anisotropic compressible hyperelastic solid will generally exhibit both volume change and distortion under hydrostatic pressure loading. In contrast, an incompressible hyperelastic solid—both isotropic and anisotropic—exhibits no change to its current state of deformation as the hydrostatic pressure is varied. Nearly incompressible hyperelastic materials are compressible, but approach an incompressible response in an appropriate limit. We examine this limiting process in the context of transverse isotropy. The issue arises as to how to implement a nearly incompressible version of a given truly incompressible material model. Here we examine how certain implementations eliminate distortion under pure pressure loading and why alternative implementations do not eliminate the distortion.     

Finite strain––isotropic hyperelasticity

This paper presents a strain energy density for isotropic hyperelastic materials. The strain energy density is decomposed into a compressible and incompressible component. The incompressible component is the same as the generalized Mooney expression while the compressible component is shown to be a function of the volume invariant J only. The strain energy density proposed is used to investigate problems involving incompressible isotropic materials such as rubber under homogeneous strain, compressible isotropic materials under high hydrostatic pressure and volume change under uniaxial tension. Comparison with experimental data is good. The formulation is also used to derive a strain energy density expression for compressible isotropic neo-Hookean materials. The constitutive relationship for the second Piola–Kirchhoff stress tensor and its physical counterpart, involves the contravariant Almansi strain tensor. The stress stretch relationship comprises of a component associated with volume constrained distortion and a hydrostatic pressure which results in volumetric dilation. An important property of this constitutive relationship is that the hydrostatic pressure component of the stress vector which is associated with volumetric dilation will have no shear component on any surface in any configuration. This same property is not true for a neo-Hookean Green’s strain–second Piola–Kirchhoff stress tensor formulation.     

聚硅氧烷硅胶的黏超弹性力学行为研究

聚硅氧烷硅胶是一类以Si——O键为主链、硅原子上直接连接有机基团的无色透明高分子聚合物, 因其具有优异的超弹性性能而广泛应用于精密减震结构、柔性电子器件等领域. 在聚硅氧烷硅胶减震结构和柔性电子器件的设计中, 材料在大变形和动态加载下的黏超弹性力学行为的精确描述至关重要. 本文针对该问题进行了系统的研究:首先, 将该硅胶的超弹性和黏弹性行为进行解耦, 确定其黏超弹性本构方程的基本框架;其次, 基于单轴拉压、平面拉伸试验确定其准静态超弹性模型的各项参数;再次, 利用霍普金森压杆冲击试验确定其黏弹性模型的各项参数;在此基础上, 将超弹性和黏弹性模型合并为适用于大应变和大应变率的黏超弹性动态本构模型;最后, 利用落锤冲击试验对该硅胶薄片的冲击变形行为进行了研究, 并利用上述建立的动态本构模型对落锤冲击过程进行了有限元模拟. 结果表明:本文建立的黏超弹性本构模型可有效预测该硅胶在冲击载荷下的力学行为, 从而为聚硅氧烷硅胶减震结构和柔性电子器件的优化设计提供了理论和应用基础.      

On constitutive modelling of porous neo-Hookean composites

In this paper a hyperelastic constitutive model is developed for neo-Hookean composites with aligned continuous cylindrical pores in the finite elasticity regime. Although the matrix is incompressible, the composite itself is compressible because of the existence of voids. For this compressible transversely isotropic material, the deformation gradient can be decomposed multiplicatively into three parts: an isochoric uniaxial deformation along the preferred direction of the material (which is identical to the direction of the cylindrical pores here); an equi-biaxial deformation on the transverse plane (the plane perpendicular to the preferred direction); and subsequent shear deformation (which includes “along-fibre” shear and transverse shear). Compared to the multiplicative decomposition used in our previous model for incompressible fibre reinforced composites [Guo, Z., Peng, X.Q., Moran, B., 2006, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J. Mech. Phys. Solids 54(9), 1952–1971], the equi-biaxial deformation is introduced to achieve the desired volume change. To estimate the strain energy function for this composite, a cylindrical composite element model is developed. Analytically exact strain distributions in the composite element model are derived for the isochoric uniaxial deformation along the preferred direction, the equi-biaxial deformation on the transverse plane, as well as the “along-fibre” shear deformation. The effective shear modulus from conventional composites theory based on the infinitesimal strain linear elasticity is extended to the present finite deformation regime to estimate the strain energy related to the transverse shear deformation, which leads to an explicit formula for the strain energy function of the composite under a general finite deformation state.