Small twist superposed on non-uniform large extensions in a circular sheet

Summary The general theory of small deformations superposed on known finite deformations of a homogeneous isotropic elastic body has been applied to find out the additional forces required to maintain the equilibrium of the final deformed state when a small twist is given to a circular sheet containing a circular hole subjected to radial forces at its outer edge.

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Zusammenfassung Auf eine Kreisringscheibe mit radialer Belastung des Außenrandes wird eine kleine Zusatzverdrehung aufgebracht. Mit Hilfe der allgemeinen Theorie kleiner Deformationen, die bekannten endlichen Verformungen in einem homogenen, isotropen elastischen Körper überlagert sind, werden die Zusatzkräfte ermittelt, die erforderlich sind, um das Gleichgewicht des endgültigen Verformungszustandes aufrechtzuerhalten.

     

任意荷载作用下层状横观各向同性弹性地基的直角坐标解

首次建立了在直角坐标系下层状地基力学问题的通用解法,改变了过去仅能在柱坐标系下进行求解此类的状况。首先将坐标系的原点选在荷载影响范围以外足够远处,从直角坐标系下的横观各向同性弹性问题的基本方程出发,利用Laplace变换及其微分性质,建立了单层横观各向同性弹性地基的状态控制方程,并利用状态空间理论给出了单层地基的解答。然后再利用传递矩阵技术,给出了任意荷载作用下的层状横观各向同性弹性地基的解析解。用提供的方法求解层状横观各向同性地基的非轴对称问题比在极坐标下求解简单、快捷。     

Elastic properties of powders during compaction. Part 2: elastic anisotropy

Isotropy in the elastic properties of powders undergoing uniaxial compaction in a cylindrical die was evaluated from in situ measurements of elastic wave speed. Shear and bulk longitudinal wave speeds were measured in both the axial (pressing) and radial directions. For the five different metal powders studied, wave speeds were generally higher in the axial direction. As such, the powder body was best described as a transversely isotropic material; complete isotropy was approached only when the powder was close to the loose packed state, or completely solid. Transversely isotropic elastic moduli analogous to the common isotropic ‘engineering’ moduli (Young’s modulus, Poisson’s ratio, etc.) were calculated by combining elastic wave speed measurements with the Saint-Venant approximation. Pseudo-isotropic elastic moduli (calculated from axial wave speed measurements and assuming elastic isotropy) were found to be only qualitatively similar to transversely isotropic elastic moduli for the axial plane.     

饱和横观各向同性分数导数黏弹性土中半封闭衬砌振动响应

在频率域内研究了饱和横观各向同性分数导数黏弹性土体中深埋圆形隧道半封闭衬砌振动响应问题。根据土体在长期沉积过程中存在各向异性的特点,将土骨架视为具有分数导数本构关系的横观各向同性黏弹性体,采用饱和多孔介质理论和弹性理论,利用衬砌内边界应力协调以及土体和衬砌界面处应力和位移连续,得到了简谐荷载作用下饱和横观各向同性黏弹性土和弹性衬砌的位移、应力和孔隙水压力解析表达式。考察了饱和经典弹性土、饱和分数导数性黏弹性土和饱和经典黏弹性土三种条件下饱和黏弹性土和衬砌各参数的影响,表明：横观各向同性面的弹性模量和衬砌厚度对系统动力响应的影响与分数导数阶数和土骨架的黏性有关;渗透系数较小时,系统存在明显的共振现象。另外,在三种条件下半封闭衬砌振动响应存在较大差异。     

Free torsional vibration of thick isotropic incompressible circular cylindrical shell subjected to uniform external pressure

The problem of torsional vibration of a cylindrical shell superposed on an initial nonhomogeneously deformed state which is caused by an uniform external pressure, is investigated. Using Green's formulation and a perturbation technique, the frequency equation for free torsional vibrations is obtained. Numerical results are presented which shows the effect of pressure.     

Diffusion of fluids through transversely isotropic solids

Summary In this paper, we investigate the problem of radial diffusion of fluids through a transversely isotropic hollow non-linearly elastic cylinder. The transversely isotropic cylinder is both sheared and stretched. We study in detail how shearing and stretching the cylinder affects the diffusion process. The influence of the anisotropy of the solid on diffusion is also determined. A comparison is made with previous work (cf. Gandhi, Rajagopal and Wineman [11]) on the radial diffusion of fluids through isotropic non-linearly elastic solids.     

Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer

The frictionless spherical indentation test is considered for a transversely isotropic elastic half-space reinforced with a thin layer whose flexural stiffness is negligible compared to its tensile stiffness. It is assumed that the deformation of the reinforcing layer can be treated as the generalized plane stress state. Closed-form analytical approximate equations for the maximum contact pressure, contact radius, and contact force are presented. The isotropic case is considered in detail.     

On free vibration of a functionally graded piezoelectric rectangular plate

Summary On the basis of three-dimensional theory equations of transversely isotropic piezoelasticity, two independent state equations with variable coefficients are derived. To this end, separation formulae for displacements and shear stresses are employed. A laminated approximation is used to transform the state equations to the ones with constant coefficients in each layer. The free vibration problem of a piezoelectric rectangular plate with a functionally graded property is then investigated. Discussion on the boundary conditions is presented.     

横观各向同性饱和土中单桩竖向振动的简化模型求解

基于饱和多孔介质理论,将土体视为横观各向同性饱和两相介质,利用分离变量法求解了横观各向同性饱和土层的竖向振动,在此基础上建立横观各向同性饱和土中单桩竖向振动的控制方程,借助桩端边界条件和三角函数的正交性对桩竖向振动问题进行了求解,并分析了相关参数对竖向振动的影响。研究表明:桩-土系统存在共振现象,复刚度的变化受激振频率的影响较大;在高频和低频时,各向异性参数对动刚度的影响不同,各向异性参数越大,复刚度的峰值越小。     

Bifurcation of a solid circular elastic cylinder under finite extension and torsion

Summary The problem of bifurcation of a solid circular cylinder subjected to finite extension and torsion is investigated using the theory of small deformations superposed on large elastic deformations. The material of the cylinder is assumed to be isotropic, elastic, homogeneous and incompressible. A numerical scheme is adopted to solve the system of partial differential equations and the associated boundary conditions governing the problem for a class of strain-energy functions. The numerical results obtained determined the critical twist corresponding to a given extension of the cylinder.     

A refined theory of transversely isotropic piezoelectric plates

Summary. A refined theory for transversely isotropic piezoelectric plates is derived from the general solution of three-dimensional transversely isotropic piezoelasticity by means of Lure operator method. As a special case, the governing differential equations for transversely isotropic elastic plates are obtained directly.     

横观各向同性功能梯度材料矩形板的自由振动

考察了沿厚度方向材料和为非均匀的功能梯度横观各向同性矩形板的自由振动。通过引入两个位移函数和两个应力函数导出了两个独立的变系数状态方程,进一步利用分层反似理论,将变系数状态方程为常纱数状态方程并求解,发现存在两类的自由振动形式：第一类对应纯板内振动,第二类对应一般的弯曲振动。给出了数值结果,讨论了材料梯度指标对固有频率的影响。     

A new approach to the finite element modelling of beams with warping effect

As a first step toward developing a finite element formulation that can model coupling among extensional, bending and torsional behaviour of beams, a new method is proposed to properly represent the warping of arbitrary cross-sections. The basic approach is to introduce a small warping displacement superimposed over flat cross-sections of a shear-flexible beam in a deformed configuration. Numerical tests involving simple isotropic beams undergoing a small elastic displacement demonstrate the validity of the new approach. The present approach can be extended to composite beams as well as isotropic beams experiencing a large deflection or finite rotation.     

Mechanical properties of FRP-FW pipes

This paper presents an analytical approach for predicting the effective elastic moduli and breaking strength of FRP-FW pipes. The FW pipe was modelled as a laminate consisting of N laminae of unidirectional fibre composite. The elastic modulus of each lamina composed of aligned, continuous transversely isotropic fibres, and an isotropic matrix was derived based on the equivalent inclusion method of Eshelby. The effective stress-strain relation of the laminate was formulated assuming the plane stress state. The breaking strength of the laminate subjected to uniaxial tension or compression was estimated by applying the quadratic failure criterion proposed by Tsai and Wu. Numerical results were compared with experimental results obtained for CFRP-FW and GFRP-FW pipes. Good agreements were obtained both in Young's modulus and in breaking strength.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

Two-dimensional static deformation of an anisotropic medium

The problem of two-dimensional static deformation of a monoclinic elastic medium has been studied using the eigenvalue method, following a Fourier transform. We have obtained expressions for displacements and stresses for the medium in the transformed domain. As an application of the above theory, the particular case of a normal line-load acting inside an orthotropic elastic half-space has been considered in detail and closed form expressions for the displacements and stresses are obtained. Further, the results for the displacements for a transversely isotropic as well as for an isotropic medium have also been derived in the closed form. The use of matrix notation is straightforward and avoids unwieldy mathematical expressions. To examine the effect of anisotropy, variations of dimensionless displacements for an orthotropic, transversely isotropic and isotropic elastic medium have been compared numerically and it is found that anisotropy affects the deformation significantly.     

Planar elastic laminates and their homogenization

Summary The state of stress and deformation of a planar elastic-homogeneous transversely isotropic thick layer in the case of plane and axisymmetric strain respectively is determined in a systematic and uniform manner using integral transforms and transfer matrices. Next, a laminate with an arbitrary number of different layers is considered without any simplifying assumptions. Then we analyze a periodic structure consisting of many thin and identical layer groups by means of a suitable homogenization, where a layer group contains two or more different transversely isotropic homogeneous basic layers. As an example exact closed form solutions for a periodically layered half space are evaluated. The well known result, that a medium which is composed of alternating thin layers of two different elastic-isotropic substances is elastostatically equivalent to a homogeneous transversely isotropic medium is extended to the above mentioned more general case. Further, the in-plane normal stresses which are discontinuous for a finite layering are evaluated in addition to the smeared ones which are continuous and correct only in the limit of a vanishing thickness of the individual layers. The explicit knowledge of the resultant elastic constants (called effective ones) turns out to be dispensable; rather, the effective material parameters (defined in this paper) which are the weighted sums of the material parameters of the basic layers are proved to be relevant. Nevertheless, for the purpose of comparison with some published results the effective elastic constants, especially for a layer group consisting of two different elastic-isotropic substances, are evaluated additionally.Dedicated to Prof. Gallus Rehm on the occasion of his 75th birthday     

Analytical layer-element approach for wave propagation of transversely isotropic pavement

Asphalt pavements have been recognised as transversely isotropic multi-layered structures. In this paper, an analytical layer-element approach is utlised to solve the wave propagation of transversely isotropic multi-layered pavement structures under the falling weight deflectometer impact load. After the application of Fourier–Hankel transform, the Navier's equation for transversely isotropic layer by impulsive force are solved analytically. The global stiffness matrix equation of multilayered structures is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Fourier–Hankel transform after the solution in the transformed domain is obtained. The layer-element of a single layer and the global stiffness matrix only contain negative exponential functions, which leads to a considerable improvement in computation ef?ciency and stability. Numerical examples are presented to demonstrate the accuracy of this method and to inversitgate the influence of the properties of transversely isotropic elastic materials on the load-displacement responses.     

Three-dimensional analysis of piezoelectric/piezomagnetic elastic media

This paper presents an exact three-dimensional method of solution for a transversely isotropic piezoelectric/piezomagnetic elastic media. The control partial differential equation set that is expressed by the elastic displacement, electric potential and magnetism potential function are established. By means of drawing two-displacement potential functions the general solution for three-dimensional transversely isotropic magneto-electro-elastic is derived. As an illustrative example, the analysis solutions of a half space body of magneto-electro-elastic acted by a point force at the origin directed along the z-axis and a point charge and a point electric current at the origin are obtained.