The invariant representation of spins with applications in the theory of finite deformations

Based on the general solution given to a kind of linear tensor equations, the spin of a symmetric tensor is derived in an invariant form. The result is applied to find the spins of the left and the right stretch tensors and the relation among different rotation rate tensors has been discussed. According to work conjugacy, the relations between Cauchy stress and the stresses conjugate to Hill's generalized strains are obtained. Particularly, the logarithmic strain, its time rate and the conjugate stress have been discussed in detail. These results are important in modeling the constitutive relations for finite deformations in continuum mechanics. The project is supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences (No. 87-52).     

极分解的一类近似计算与比较

基于级数展开给出了极分解中右伸长张量 的级数表示,通过对级数的项的选取得到右伸长张量的不同近似表达式。针对不同级数展开表示,得到表达式最小误差的级数展开形式。进而结合一些简单实例,验证误差了近似公式的有效性。最后与黄模佳等关于计算右伸长张量 和转动张量 的近似表达式进行了比较,本文的级数展开方式得到的右伸长张量 和转动张量 的近似表达式不但简洁,而且计算精度更高、适用范围更广。     

On the derivative of the square root of a tensor and Guo's rate theorems

Formulas for the derivative of the square root of a positive definite, symmetric, second-order tensor are derived and used to obtain expressions for the material time derivatives of the right and left stretch tensors.     

Recent investigations on strain and stress rates in nonlinear continuum mechanics

Presenting some recent considerations and results, the present paper deals with two basic concepts of the continuum mechanics: strain-and stress-rates. Upon a brief systematic survey of concepts of strain and stress, a so far unknown explicit expression for the rate of the right stretch tensor is offered in absolute notation. This paper then suggests to distinguish two ways of defining objective stress-rates. Following the second procedure, after analyzing several particular cases, the author proposes a generalized Jaumann flux, which contains the majority of the existing definitions for stress-rate and the Hill’s result as well.     

On the Derivative of Some Tensor-Valued Functions

An explicit expression of the derivative of the square root of a tensor is provided, by using the expressions of the derivatives of the eigenvalues and eigenvectors of a symmetric tensor. Starting from this result, the derivatives of the right and left stretch tensor U, V and of the rotation R with respect to the deformation gradient F, are calculated. Expressions for the material time derivatives of U, V and R are also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some basis-free expressions for stresses conjugate to Hill’s strains through solving the tensor equation AX + XA = C

In the present paper, some new and compact basis-free expressions are derived for representing stresses conjugate to the Lagrangean Hill’s strain measures. Solving a tensor equation in the form of AX + XA = C, a rather simple general expression is obtained for the conjugate stresses in the case of distinct principal stretches. The result decomposes the conjugate stresses additively into two parts, such that one part is coaxial with the right stretch tensor while another part is orthogonal to it. An expression for the case of double-coalescent principal stretches is also obtained. As an example of the general results, simple expressions are obtained for the stress conjugate to the logarithmic strain.     

组合超弹性材料中的空穴生成与增长

本文研究了一种组合不可压超弹性材料圆柱体中空穴的生成与增长问题，得到了这种材料受表面均布拉伸死荷载和轴向拉压共同作用下空穴生成问题的解析解，得到了不同组合情况下圆柱体中空穴生成时的临界载荷及分叉曲线，发现组合材料可以发生右分叉，也可以发生左分叉；给出了空穴生成后的应力分布，并讨论了所存在的应力间断和应力集中问题；通过能量比较分析了解的稳定性，讨论了发生右分叉或左分叉的条件，并分析了材料中预存微孔的增长情况。     

单轴拉伸下液晶高弹体力学行为的热力序耦合特性

本文利用液晶高弹体的应力－应变本构方程和力－序耦合方程,研究了沿指向矢单轴拉伸下液晶高弹体力学行为的热力序耦合特性。由于力－序耦合,有序度在拉伸作用下变大,从而影响了液晶高弹体的应力－应变关系,使得应力在相同伸长下变小。不同温度下应力受到有序度变化影响的程度不同.由于有序度随温度升高而减小,当工程应力不变时,伸长随着温度升高而减小;当伸长不变时,应力随着温度升高而增大;伸长随温度的变化关系和应力随温度的变化关系都呈非线性。     

自旋张量的绝对表示及其在有限变形理论中的应用

基于对一类线性张量方程的一般解法,导出了任一对称张量所对应的自旋张量的绝对表示。该结果可以很自然地用于研究左和右伸长张量的自旋并研讨在连续介质力学中常见到的各种转动率张量间的关系。一个重要的公式,即Hill意义下广义应变的共轭应力和Cauchy应力之间的关系,从功共轭原理建立了起来。尤其是详细讨论了对数应变的时间变率及相应的共轭应力。无疑,上述结果对有限变形条件下本构理论的研究是颇为重要的。     

Volume changes associated with the deformation of rubber-like solids

Volumechanges accompanying the deformation of rubber-like solids are analysed on the basis of isotropic elasticity theory. In particular, a simple, but general, result relating the volume and the stretch in simple tension is obtained. This is achieved by the introduction of certain modified principal stretches which allow the dependence of the strain energy on the isochoric and the dilatational parts of the deformation to be considered separately. The fact that volume changes in rubber-like solids are typically of order 0.01 % is used to linearize the stress-deformation relations in the dilatation. This enables the dilatation to be given explicitly as a function of the stretch in simple tension. Specific results are obtained for certain classes of constitutive law and good agreement with the experimental data for simple tension is demonstrated.Results for equi-biaxial tension and pure shear are also given in anticipation of further experimental data becoming available. The need for volume-change data for a wide variety of types of strains is emphasized.     

Critical stretch for formation of a cylindrical void in a compressible hyperelastic material

Formation of a cylindrical void in an infinitely long compressible hyperelastic cylinder under axial and radial stretches is examined. The cavitation phenomenon is viewed here as a bifurcation of a solution with a cavity from the homogeneously deformed configuration, taking place when the applied radial stretch reaches a certain critical value. This amounts to modeling the underlying phenomenon as a kind of elastic instability. A formulation of shooting-method type is presented to derive an equation which gives the critical radial stretch for a prescribed axial stretch. For a special class of hyperelastic solids, called modified Blatz-Ko material, the obtained equation leads to an explicit expression for the critical stretches and stresses. Some analytical as well as numerical calculations are carried out to explicitly obtain the critical values for cavitation. The results are summarized in the form of cavitation curves in the two-dimensional space of axial and radial stretches or stresses. Influence of the axial stretch on the critical radial stretch is discussed. Throughout the paper, the corresponding results for a spherically symmetric void formation are referred to when appropriate and compared with the cylindrical case of the present interest. It is then indicated that in the state of equitriaxial stretch, cavitation into a cylindrical shape is likely to occur at lower stretch than into a spherical one.     

Cavity formation at the center of a sphere composed of two compressible hyper-elastic materials

IntroductionHorgan[1] reviewedthecavitatedbifurcationproblemforhyper_elasticmaterials,includinginhomogeneousandanisotropicmaterialsaswellashomogeneousandisotropicmaterials .Forincompressiblematerials,HorganandPence[2 ,3 ] examinedtheeffectofmaterialinhomogeneityontheformationandgrowthofvoidandobtainedananalyticsolutionofthecavitatedbifurcationproblemforasolidspherecomposedoftwoneo_Hookeanmaterials.Thebifurcationmayoccurnotonlytotherightbutalsototheleftforthecomposedsphere .Thestabilitiesofth…     

Some specific features and consequences of the thermal response of rubber under cyclic mechanical loading

The present paper deals with the specificities of the thermal response of rubber under cyclic mechanical loading at constant ambient temperature. This question is important, since the stabilized thermal response is used in fatigue life criteria, especially for the fast evaluation of fatigue life. For this purpose, entropic coupling in a thermo-hyperelastic framework is first used to predict the variation in the heat source produced or absorbed by the material during cyclic loading. The heat diffusion equation is then used to deduce temperature variations under adiabatic and non-adiabatic conditions. The influence of several parameters on the stabilized thermal response is studied: signal shape, frequency, minimum and maximum stretch levels, multiaxiality of the mechanical state. The results show that, in the steady-state regime, the mean value between the maximum and minimum temperature variations over a mechanical cycle is different from zero. This is due to the specific variation in the heat source, which depends on both the stretch rate and the stretch level. This result has numerous consequences, in particular for fatigue. Indeed, the stabilized mean value between the maximum and minimum temperature variations during fatigue tests does not reflect only fatigue damage, since the entropic coupling also leads to a value different from zero. This is a major difference with respect to materials exhibiting only isentropic coupling, such as metallic materials.     

Physical invariants for nonlinear orthotropic solids

Seven invariants, with immediate physical interpretation, are proposed for the strain energy function of nonlinear orthotropic elastic solids. Three of the seven invariants are the principal stretch ratios and the other four are squares of the dot product between the two preferred directions and two principal directions of the right stretch tensor. A strain energy function, expressed in terms of these invariants, has a symmetrical property almost similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress–strain relations are given. Using principal axes techniques, the formulation is applied, with mathematical simplicity, to several types of deformations. In simple shear, a necessary and sufficient condition is given for Poynting relation and two novel deformation-dependent universal relations are formulated. Using series expansions and the symmetrical property, the proposed general strain energy function is refined to a particular general form. A type of strain energy function, where the ground state constants are written explicitly, is proposed. Some advantages of this type of function are indicated. An experimental advantage is demonstrated by showing a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed.     

An Average-Stretch Full-Network Model for Rubber Elasticity

Two constitutive models that are based on the classical non-Gaussian, Kuhn-Grün probability distribution function are reviewed. It is shown that all chains of a network cell structure comprised of a finite number of identical chains in an affine deformation referred to principal axes may have the same invariant stretch, if and only if the chains are oriented initially along any of eight directions forming the diagonals of a unit cube. The 4-chain tetrahedral and the 8-chain cubic cell structures are familiar admissible models having this property. An easy derivation of the constitutive equation for the Wu and van der Giessen full-network model of initially identical chains arbitrarily oriented in the undeformed state is presented. The constitutive equations for the neo-Hookean model, the 3 -chain model, and the equivalent 4- and 8-chain models are then derived from the Wu and van der Giessen equation. The squared chain stretch of an arbitrarily directed chain averaged over a unit sphere surrounding all chains radiating from a cross-link junction as its center is determined. An average-stretch, full-network constitutive equation is then derived by approximation of the Wu and van der Giessen equation. This result, though more general in that no special chain cell morphology is introduced, is the same as the constitutive equation for the 4- and 8-chain models. Some concluding remarks on extensions to amended models are presented.     

A closed form solution for the uniaxial tension test of biological soft tissues

This paper concerns the modeling of biological soft tissues in the framework of anisotropic hyperelasticity. A closed form solution is proposed in the special case where uniaxial unconstrained tension loading is applied to a strip modeled by Holzapfel-Gasser-Ogden's (HGO) strain energy. The classical Cardano's formula is used to calculate the solution. This solution is investigated for different values of β which represents the angle between the collagen fibers and the circumferential direction. This study allows for the understanding of the reason why a one-to-one correspondence does not exist between the principal stretch and the fourth invariant of the right Cauchy-Green deformation tensor. It is demonstrated that the relationship becomes non-bijective when β is greater than a critical angle of 54.73^°. The HGO model is implemented into an in-house finite element program and numerical results are in good agreement with theoretical ones.     

Fiber Remodeling During Torsion of a Fiber Reinforced Hyperelastic Cylinder��Unloading Behavior

Previous studies introduced a constitutive theory for fiber reinforced hyperelastic materials that allows the fibers to undergo microstructural changes. In this theory, increasing deformation of the matrix leads to increasing stretch of the fibers that causes their gradual dissolution. The dissolving fibers reassemble in the direction of maximum principal stretch of the matrix. The implications of the constitutive theory were first studied for two homogeneous deformations: uniaxial extension along the fibers and simple shear in the direction normal to the fibers. The constitutive theory was then used in treatment of the non-homogeneous deformation of combined axial stretch and twisting. The emphasis was on the determination of the influence of increasing axial stretch and twist on the spatial distribution of fiber dissolution and reassembly within the cylinder and also on the axial force and torque applied to the end faces of the cylinder. The present work is concerned with another aspect of combined axial stretch and twisting of the cylinder, namely unloading following dissolution and reassembly of some of the fibers. In this case, the cylinder is given an initial twist until there is an inner core of original fiber/matrix material and an outer sheath of remodeled fiber/matrix material. A condition is established that determines the combinations of axial stretch and twist that cause no additional dissolution and reassembly of fibers during unloading. It is also shown that there is a residual axial stretch and twist if the axial force and torque become zero. A numerical example illustrates this for a particular choice of matrix and fiber properties.     

Axonal Buckling Following Stretch Injury

Diffuse brain injury is caused by rapid rotation of the head, and causes strain injury to tissue throughout the brain. Following strain injury, axons exhibit delayed recovery, showing regional buckling behavior immediately after stretch and returning to their original appearance over an extended period of time. This axonal buckling is hypothesized to occur as a result of localized stretching within the axon: Rapid strain causes mechanical damage to microtubules, increasing the effective length of axons. This damage is repaired gradually returning the axon to its initial length.Here, we test the hypothesis that localized stretching is a possible explanation for the regional buckling behavior. An elongated region of axon is modeled as an Euler beam on an elastic foundation, where the foundation represents the surrounding brain tissue, which consists of glial cells and extracellular matrix. After stretch the elastic foundation returns immediately to its pre-stretch length, while the axon is initially elongated and returns to its original length over a longer period of time. The model exhibits solutions similar to those observed experimentally in post-stretch axons, with undulations that have a similar wavelength and amplitude.     

Thermomechanical modeling of the thermo-order-mechanical coupling behaviors in liquid crystal elastomers

Liquid crystal elastomer is a kind of anisotropic polymeric material, with complicated micro-structures and thermo-order-mechanical coupling behaviors. In this paper, we propose a method to systematically model these coupling behaviors. We derive the constitutive model in full tensor structure according to the Clausius-Duhem inequality. Two of the constitutive equations represent the mechanical equilibrium and the other two represent the phase equilibrium. Choosing the total free energy as the combination of the neo-classical free energy and the Landau-de Gennes nematic free energy, we obtain the Cauchy stress-deformation gradient relation and the order-mechanical coupling equations. We find the analytical homogeneous solutions of the deformation for the typical mechanical loadings, such as uniaxial stretch, and simple shear in any directions. We also compare the compression behavior of prolate liquid crystal elastomers with the stretch behavior of oblate liquid crystal elastomers. As a result, the stress, strain, temperature, order parameter, biaxiality and the direction of the director of liquid crystal elastomers couple with each other. When the prolate liquid crystal elastomer sample is stretched in the direction parallel to its director, the deviatoric stress makes the mesogens more order and increase the transition temperature. When the sample is sheared or stretched in the direction non-parallel to the director, the director of the liquid crystal elastomer will rotate, and the biaxiality will be induced. Because of the order-mechanical coupling, under infinitesimal deformation, liquid crystal elastomer has anisotropic Young’s modulus and zero shear modulus in the direction parallel or perpendicular to the director. While for the oblate liquid crystal elastomers, the stretch parallel to the director will cause the rotation of the director and induce the biaxiality.     

边界受限下的轴向压缩超弹性圆管的失稳分析

橡胶制品广泛应用于社会生产和日常生活。本文主要研究外边界受限的超弹性圆管受轴向压缩时的稳定性问题,在非线性弹性理论框架下推导控制方程,对其进行线性分岔分析。考虑沿轴向对称分布的失稳模态,得到临界模态以及对应的临界主伸长,同时刻画出临界主伸长以及临界应力与壁厚之间的关系。结果表明,圆管壁越薄,发生分岔时的临界主伸长更小,更不容易发生失稳。此外,结合封隔器的密封原理,推导出封隔器圆筒在进行油液密封时轴向应力与接触应力的表达式,修正了此前的推导结果,为封隔器在密封性能的判断上提供理论依据。