翻转后的不可压缩neo-Hookean圆柱管的有限变形

研究了由径向横观各向同性不可压缩的neo-Hookean材料组成的圆柱形管在翻转后的有限变形问题。利用材料的不可压缩条件和半逆解法对相应的数学模型进行求解,并根据边界条件得到了翻转后的圆柱形管的内半径以及轴向伸长率应满足的非线性方程组。通过数值算例讨论了材料参数和结构参数对翻转后圆柱形管的内半径以及轴向伸长率变化的影响。结果表明：初始厚度对翻转后圆柱形管的内半径与轴向伸长率没有本质上的影响;而径向各向异性参数却有本质上的影响,特别是在轴向伸长率方面。     

Torsional Swelling of a Hyperelastic Tube with Helically Wound Reinforcement

This paper examines the combination of radial deformation with torsion for a circular cylindrical tube composed of a transversely isotropic hyperelastic material subject to finite deformation swelling. The stored energy function involves separate matrix and fiber contributions such that the fiber contribution is minimized when the fiber direction is at a natural length. This natural length is not affected by the swelling. Hence swelling preferentially expands directions that are orthogonal to the fiber. The swelling itself is described via a swelling field that prescribes the local free volume at each location in the body. Such a treatment is a relatively simple generalization of the conventional incompressible theory. The direction of transverse isotropy associated with the fiber reinforcement is described by a helical winding about the tube axis. The swelling induced preferential expansion orthogonal to this direction induces the torsional aspect of the deformation. For a specific class of strain energy functions we find that the twist increases with swelling and approaches a limiting asymptotic value as the swelling becomes large. The fibers reorient such that fibers at the inner portion of the tube assume a more circumferential orientation whereas, at least for small and moderate swelling, the fibers in the outer portion of the tube assume a more axial orientation. For large swelling the fibers in the outer portion of the tube reorient beyond the axial orientation, and so are described by helices with orientation in the opposite sense to that in the reference configuration.      

Extension,inflation and torsion of a residually stressed circular cylindrical tube

In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.     

On Structured Surfaces with Defects: Geometry,Strain Incompatibility,Stress Field,and Natural Shapes

In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.     

Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres

The effects of material anisotropy and inhomogeneity on void nucleation and growth in incompressible anisotropic nonlinearly elastic solids are examined. A bifurcation problem is considered for a composite sphere composed of two arbitrary homogeneous incompressible nonlinearly elastic materials which are transversely isotropic about the radial direction, and perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Several types of bifurcation are found to occur. Explicit conditions determining the type of bifurcation are established for the general transversely isotropic composite sphere. In particular, if each phase is described by an explicit material model which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials, phenomena which were not observed for the homogeneous anisotropic sphere nor for the composite neo-Hookean sphere may occur. The stress distribution as well as the possible role of cavitation in preventing interface debonding are also examined for the general composite sphere.     

Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

横观各向同性圆球壳的自由扭转振动

本文用弹性动力学理论研究横观各向同性圆球壳的轴对称自由扭转振动问题,求出位移和应力的解析表达式,揭示了壳体在子午线方向和半径方向的耦合振动特性,文末算例给出不同几何尺寸和材料性质圆球壳固有频率和振型的数字计算结果.     

Stress field of orthotropic cylinder subjected to axial compression

Based on the constancy hypothesis of material volume, the circumferential and radial stresses of a cylinder specimen are analyzed when the cylinder is subject to a loading along the axial direction. The circumferential and radial stress distribution is a power function of radius parameter when the constitutive relation of specimen material is orthotropic. The stress distribution is a quadratic function of radius parameter for transversely isotropic material. Along the cylinder axial line, the circumferential and radial stresses are maximum and equal to each other. In the circumference boundary surface, the radial stress is zero and the circumferential stress value is minimal. The failure theory of maximum tensile circumferential strain is applied to calculate the critical axial loading. The circumference-boundary-layer failure criterion of orthotropic cylinders is described with the Hill-Tsai strength theory. The obtained strength theory is related to axial stress and mechanical properties of specimen material and to the specimen axialdeformation strain rate and the change rate of strain rate.     

Wavy-edged fractures in axially split aluminium tubes

This paper is motivated by the somewhat unusual need to gain insight into the phenomenology of the mechanism by which a wavy edge is formed on the wreckage of some aircraft fuselage skins associated with aircraft destroyed in flight by on-board explosion.In order to explore the role of the plastic zone adjacent to the crack tip whilst avoiding the practical complications of generating the fractures explosively, simple quasi-static experiments have been carried out on aluminium tubes. Oversized rigid dies were pushed inside the tubes along their axes to generate fractures in Mode I and Mode III. It is conjectured that wavy edges are associated with fractures resulting from internal expansion of the tube by a travelling, internal, radial ring pressure region. The pressurised region behind the crack tip would be produced by explosively generated internal pressure being vented at the crack and, for the purpose of this study, is considered to be equivalent to that generated by the die. The production of such cracks is clearly demonstrated experimentally and contrasted with the plain-edge fractures produces during Mode III tearing fracture.A damage-model-based finite element analysis has been conducted to simulate the propagation of the crack and provide further insight into the strain and stress fields along the fractured edges. Both the experimental and numerical results show that this particular type of ring loading has to be applied to the tube to produce the wavy edge. Such a load expands the fractured flaps in the radial direction, stretching the material in the circumferential direction and, crucially, in the axial direction. The latter generates a relatively wide plastic wake close to and parallel to the fracture edge as the tube fails within which axial plastic strain predominates. Constrained by the remaining part of the tube that has not undergone plastic deformation, sufficient axial residual compressive stress can be produced in the plastic wake to produce a wavy edge which results from local buckling in the plastic wake. This mechanism suggests that ripples observed on the edges of fuselage skin wreckage are possible signatures of an internal explosion. The work described herein is also relevant to the deformation in a failed high-pressure gas pipe following the propagation of a ductile crack as noted previously in the literature.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.     

Finite deformation swelling of a temperature-sensitive hydrogel cylinder under combined extension-torsion

The swelling behavior of a temperature-sensitive poly-N-isopropylacrylamide(PNIPAM) hydrogel circular cylinder is studied subjected to combined extension-torsion and varied temperature. In this regard, a semi-analytical solution is proposed for general combined loading. A finite element(FE) analysis is conducted, subjecting a hydrogel cylinder to the combined extension-torsion and the varied temperature to evaluate the validity and accuracy of the solution. A user-defined UHYPER subroutine is developed and verified under free and constrained swelling conditions. The FE results illustrate excellent agreement with the semi-analytical solution. Due to the complexity of the problem, some compositions and applied loading factors are analyzed. It is revealed that for larger cross-linked density and larger ending temperature, the cylinder yields higher stresses and smaller radial swelling deformation. Besides, the radial and hoop stresses increase by applying larger twist and axial stretch. The hoop stresses intersect at approximately R/Rout = 0.58, where the hoop stress vanishes. Besides, the axial force has direct and inverse relationships with the axial stretch and the twist, respectively. However, the resultant torsional moment behavior is complex, and the position of the maximum point varies significantly by altering the axial stretch and the twist.     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

热超弹性圆筒的不稳定性

应用有限变形弹性理论分析了受内压和轴向拉伸作用的不可压热超弹性圆筒发生非均 匀变形的不稳定性问题. 受内压和轴向拉力作用的薄壁圆筒，当内压较小时，圆筒发生稳定 的均匀膨胀变形；当内压大于某一临界值时，圆筒产生复杂的非均匀变形，其一部分膨胀变 形很大，形如``灯泡'状，而另一部分仅仅是轻微膨胀，且此时的变形是不稳定的. 但对厚 壁圆筒而言，不论压力如何，总是发生稳定的均匀膨胀变形. 根据圆筒的变形曲线，给出了 圆筒可以发生不稳定变形的临界厚度. 同时，讨论了轴向拉伸和温度场对圆筒变形的影响.     

考虑微观变形特征的水凝胶均匀和非均匀溶胀分析及其影响参数研究

本文给出考虑微观变形的水凝胶溶胀的分析模型,该模型假设构成聚合物网络的单链受到由于周围链的作用而产生类似圆管状的约束,并且认为每个单链变形与网络变形之间存在非仿射关系.利用该模型分析了凝胶在自由溶胀情形,预拉伸凝胶单一方向溶胀情形以及具有刚性核的球形凝胶溶胀至平衡状态情形的变形特征.研究表明, 对于自由溶胀的均匀变形情形, 水凝胶微观单链与宏观网络的变形相一致,伸长率随链段数的增加而增大, 随单链密度的增大而减小.伸长率也会随有效圆管几何参数数值的增大而增大,但是当外部溶剂压力达到一定数值以后单链的圆管约束对凝胶溶胀变形影响逐渐减小.在单一方向溶胀状态下,预拉伸凝胶会出现微观单链和宏观网络伸长率相等的溶胀平衡状态,此时约束应力等于零.具有刚性核的球形凝胶溶胀至平衡状态时产生径向和环向伸长率不相等的非均匀球对称变形.在刚性核的附近, 单链和网络的径向伸长率均大于自由溶胀状态伸长率.在远离刚性核的位置, 单链和网络的伸长率均接近自由溶胀状态的伸长率.网络渗透压随有效圆管几何参数数值的增大而降低,溶剂分子体积分数则随有效圆管几何参数数值的增大而增大.所建立的分析模型可以模拟凝胶溶胀导致的微观单链变形.      

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: I. Mechanical Equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.     

Transient torsional wave propagation in a transversely isotropic tube

Summary By use of the separation of variables method and the Laplace transformation, the two-dimensional transient torsional wave propagation problem in a transversely isotropic tube is studied when a torque is suddenly applied to its end surface. The results show that, for the discontinuous distribution of the impact shear stress, the region of the 2D stress distribution in a transversely isotropic tube becomes large with the increase of the anisotropy of the material. Received 13 June 1997; accepted for publication 17 June 1998     

粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析

研究了上下表面粘贴压电层的功能梯度材料Timoshenko梁在升温及电场作用下的过屈曲行为。在精确考虑轴线伸长和一阶横向剪切变形的基础上,建立了压电功能梯度Timoshenko层合梁在热-电-机械载荷作用下的几何非线性控制方程。其中,假设功能梯度的材料性质沿厚度方向按照幂函数连续变化,压电层为各向同性均匀材料。采用打靶法数值求解所得强非线性边值问题,获得了在均匀电场和横向非均匀升温场内两端固定Timoshenko梁的静态非线性屈曲和过屈曲数值解。并给出了梁的变形随热、电载荷及材料梯度参数变化的特性曲线。结果表明,通过施加电压在压电层产生拉应力可以有效地提高梁的热屈曲临界载荷,延缓热过屈曲发生。由于材料在横向的非均匀性,即使在均匀升温和均匀电场作用下,也会产生拉-弯耦合效应。但是对于两端固定的压电-功能梯度材料梁,在横向非均匀升温下过屈曲变形仍然是分叉形的。     

饱和多孔弹性Timoshenko悬臂梁的拟静力弯曲

在经典单相Timoshenko梁变形和孔隙流体仅沿多孔梁轴向运动的假定下,基于不可压饱和多孔介质的三维理论,本文首先建立了横观各向同性饱和多孔弹性Timoshenko悬臂梁拟静力弯曲的一维数学模型,并给出了相应的边界条件。其次,利用Laplace变换及其数值逆变换,分析了端部不同渗透条件下,饱和多孔弹性Timoshenko悬臂梁在端部梯载荷作用下的拟静力响应,给出了饱和多孔Timoshenko悬臂梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应曲线,并与饱和多孔Euler-Bernoulli悬臂梁的响应进行了比较,考察了梁长细比对弯曲的影响。数值结果表明：固相骨架与孔隙流体的相互作用具有粘性效应,梁弯曲的拟静态挠度具有蠕变行为,端部渗透条件对梁的弯曲响应有显著的影响,并且,饱和多孔弹性Timoshenko悬臂梁的拟静态响应亦存在Mandel-Cryer现象。