Some analytical solutions for Saint-Venant torsion of non-homogeneous anisotropic cylindrical bars

The Saint-Venant torsion of linearly elastic anisotropic cylindrical bars with solid and hollow cross-section is treated. The shear flexibility moduli of the non-homogeneous bar are given functions of the Prandtl's stress function of considered cylindrical bar when its material is homogeneous. The solution of the torsion problem of non-homogeneous anisotropic bar is expressed in terms of the torsion and Prandtl's stress functions of the corresponding homogeneous anisotropic bar having the same cross-section as the non-homogeneous bar.     

Prandtl’s formulation for the Saint–Venant’s torsion of homogeneous piezoelectric beams

The Saint–Venant torsional problem for homogeneous, monoclinic piezoelectric beams is formulated in terms of Prandtl’s stress function and electric displacement potential function. The analytical approach presented in this paper generalizes the known formulation of Prandtl’s solution which refers to homogeneous elastic beams. The Prandtl’s stress function and electric displacement potential function satisfy the so called coupled Dirichlet problem (CDP) in the cross-sectional domain. A direct and a variational formulation are developed. Exact analytical solutions for solid elliptical cross-section and hollow circular cross-section and an approximate solution based on a variational formulation for thin-walled closed cross-section are presented.     

Torsin of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elastic equations for a cylindrical section of a tree

Considering a cylindrical section of a tree subjected to loads independent of x₃ as a relaxed Saint-Venant's problem, it was shown that plane sections remain plane. Since plane sections remain plane, the displacement equations for the neutral fiber derived using either the relaxed Saint-Venant's problem or elementary beam theory are equivalent. The stresses in the plane of the transverse cross-section were found to equal to zero. Therefore, it is appropriate to use elementary beam theory to estimate the three-dimensional stress functions when the wood is considered to be homogeneous. In addition the three-dimensional displacement equations allow the required elastic coefficients in cylindrical coordinates to be measured from full size samples.     

Saint-Venant torsion analysis of bars with rectangular cross-section and effective coating layers

This paper investigates the torsion analysis of coated bars with a rectangular cross-section. Two opposite faces of a bar are coated by two isotropic layers with different materials of the original substrate that are perfectly bonded to the bar. With the Saint-Venant torsion theory, the governing equation of the problem in terms of the warping function is established and solved using the finite Fourier cosine transform. The state of stress on the cross-section, warping of the cross-section, and torsional rigidity of the bar are evaluated. Effects of thickness of the coating layers and material properties on these quantities are investigated. A set of graphs are provided that can be used to determine the coating thicknesses and material properties so as to keep the maximum von Mises stress on the cross-section below an allowable value for effective use of the coating layer.     

Second-order solution of Saint-Venant's problem for an elastic bar predeformed in flexure

We use the method of Signorini's expansion to analyze the Saint-Venant problem for an isotropic and homogeneous second-order elastic prismatic bar predeformed by an infinitesimal amount in flexure. The centroid of one end face of the bar is rigidly clamped. The complete solution of the problem is expressed in terms of ten functions. For a general cross-section, explicit expressions for most of these functions are given; the remaining functions are solutions of well-posed plane elliptic problems. However, for a bar of circular cross-section, all of these functions are evaluated and a closed form solution of the 2nd-order problem is given. The solution contains six constants which characterize the second-order flexure, bending, torsion and extension of the bar. It is found that when the total axial force vanishes, the second-order axial deformation is not zero; it represents a generalized Poynting effect. The second-order elasticities affect only the second-order axial force.     

Extremum principles for certain hyperelastic materials

A hyperelastic material is here said to be of class H_m if the elastic potential is a homogeneous function of order m + 1 in the components of the Lagrangian displacement gradient. It is shown that a single solution to a boundary value problem generates an infinite family of solutions to a family of related boundary value problems. Assuming that a solution to a boundary value problem exists, it is shown that it is unique provided that the material is stable in the sense of Hill in a deleted neighbourhood of the stress-free state. A minimum theorem concerning the strain energy and the virtual work of the prescribed forces is established for the equilibrium configurations, and a maximum theorem concerning the virtual work of the prescribed surface displacements and the complementary stress energy is established for compatible stress fields. As an application, upper and lower bounds are found for the torsional stiffness of a cylindrical bar of square cross section under infinitesimal deformation.     

The vibration and buckling of laminated non-homogeneous orthotropic conical shells subjected to external pressure

In this paper, the free vibration and buckling of laminated homogeneous and non-homogeneous orthotropic truncated conical shells under lateral and hydrostatic pressures are studied. At first, the basic relations, the modified Donnell type dynamic stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the Young's moduli and density of which vary piecewise continuously in the thickness direction. Applying superposition and Galerkin methods to the foregoing equations, the buckling pressures and dimensionless frequency parameter of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the effects of the number and ordering of layers, the variations of conical shell characteristics, together and separately variations of the Young's moduli and densities of the materials of layers on the critical lateral and hydrostatic pressures, and frequency parameter are found for different mode numbers. The results are compared with other works.     

Thermal gradient effect on the free vibration of a cardioid cross-section cylindrical shell

For the first time, a new cross-section profile and efficient method are developed for the vibration analysis of isotropic and orthotropic cylindrical shells having circumferentially varying profile of a cardioid cross-section expressed as an arbitrary function, under thermal gradient effect. The governing equations of orthotropic cylindrical shells with varying thermal gradient around its circumference are derived as a boundary-value problem and solved numerically as an initial-value problem, based on the framework of Flügge's shell theory, transfer matrix approach and Romberg integration method. As a semi-analytical procedure, the trigonometric functions are used with Fourier's approach to approximate the solution in the longitudinal direction and also to reduce the two-dimensional problem to one-dimensional one. The thermal gradient is assumed to arise due to the variation of Young's moduli and shear modulus, along the circumferential direction of the shell. The results are obtained to indicate the effects of cardioid cross-section on the natural frequencies and corresponding mode shapes in the thermal environment as well as the sensitivity of the vibration behavior to the thermal gradient ratio and the orthotropy of the shell is also investigated for different types of vibration modes. In general, close agreement between the obtained results and those of other researchers has been found.     

Comparison of analytical techniques employed to reduce the data obtained for steel samples subjected to high ambient pressure and axial tension

This report reexamines some of the experimental data collected by P.W. Bridgman and published in his book ‘Studies in Large Plastic Flow and Fracture’. Of special concern, for the present considerations, is the problem of the flow and resultant ‘necking’ which occurs when a cylindrical bar of steel is subjected to high ambient pressures and axial tension. Bridgman developed an equation which corrected the average stress across the minimum cross-section of the ‘necked’ portion of the sample for the effects of plastic stress concentration. A subsequent analysis by one of the present authors offered another relation for the stress concentration which provided a more complete theoretical analysis of the problem. This report documents the statistical evidence that the new procedure provides a superior correlation to Bridgman's data than Bridgman's original analysis.     

Stress Analysis for an Elastic Semispace with Surface and Graded Layer Coatings under Induced Torsion

In this paper, the axisymmetric torsional problem of a coating structure consisting of a surface coating, a functionally graded layer and a substrate under a rigid cylindrical punch is investigated. The coating and substrate are homogeneous materials with distinct physical properties while the intermediate layer is inhomogeneous with its shear modulus changing exponentially along the thickness direction. The Hankel integral transform technique is employed to reduce the torsional problem to a singular integral equation with a Cauchy kernel. The circumferential shear stress and displacement fields in the coating structure are calculated by solving the integral equation numerically. The results show that the stiffness ratio has significant effect on the distribution of the circumferential stress and displacement at the interface.     

Torsional deformations in incompressible fibre-reinforced cylindrical pipes

The first part of the paper deals with an extension of the classical Rivlin's solution of the torsion problem of a neo-Hookean pipe. The second part concerns a study of the passive torsional deformation processes in a fibre-reinforced cylindrical dummy of the beating heart. Especially, the dependence of the torsional and volumetric stiffness of the cylindrical pipe on different geometric and material parameters is discussed through a set of numerical simulations.     

On non-local and non-homogeneous elastic continua

A thermodynamic framework endowed with the concept of non-locality residual is adopted to derive non-local models of integral-type for non-homogeneous linear elastic materials. Two expressions of the free energy are considered: the former yields a one-component non-local stress, the latter leads to a two-component local–non-local stress since the stress is expressed as the sum of the classical local stress and of a non-local component identically vanishing in the case of constant strains. The attenuation effects are accounted for by a symmetric space weight function which guarantees the constant strain requirement as well as the dual constant stress condition everywhere in the body. The non-local and non-homogeneous elastic structural boundary-value problem under quasi-static loads is addressed in a geometrically linear range. The complete set of variational formulations for the structural problem is then provided in a unitary framework. The solution uniqueness of the non-local structural model is proved and the non-local FEM is addressed starting from the non-local counterpart of the total potential energy. Numerical applications are provided with reference to a non-homogeneous bar in tension using the Fredholm integral equation and the non-local FEM. The solutions show no pathological features such as numerical instability and mesh sensitivity for degraded bar conditions.     

Generalized Poynting Effects in Predeformed Prismatic Bars

We use the Signorini expansion method to determine second-order Saint-Venant solution for an infinitesimally bent and stretched bar. The bar in the unstressed reference configuration is straight, prismatic, isotropic, homogeneous and made of a second-order elastic material. These solutions and those found earlier for a pretwisted bar give generalized Poynting effects. A bar when bent stretches and the elongation is determined by the first and second-order elasticities, area of cross-section, torsional rigidity, bending vector and the inertia tensor. When an infinitesimally twisted bar is deformed, there is a second-order bending deformation even when there is no resultant bending moment applied on the end faces. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

A Universal Expression for Bounds on the Torsional Rigidity of Cylindrical Shafts Containing Fibers with Arbitrary Transverse Cross-Sections

We derive upper and lower bounds for the torsional rigidity of host shafts containing a number of cylindrical fibers. The transverse cross-sections of the host shaft and the fibers are simply connected, but could be arbitrary in shape. Utilizing the fact that the torsion solution of a homogeneous host shaft with simply connected cross-section can be known, we propose a method to construct statically and kinematically admissible fields interior to the matrix and to the fibers. Previous developments on bounding the torsional rigidity of composite shaft so far are confined to circular fibers. Here we try to simulate fibers with non-circular cross-section and incorporate the interactions of the cross-sectional shapes of the host shaft and the fibers at the same time. Proceeding from extremal principles of elasticity, together with propositions of some domain integration procedures, we provide a universal expression for bounds on the torsional rigidity of the composite shaft. The exact expressions depend on the constituent information of the fibers and the host shaft, which could offer useful information to tailor the shape and the arrangement of the constituents to achieve an optimal value.     

An experimental study on the static and impact torsional buckling of cylindrical shells

In this paper, experimental studies are carried out on the buckling of circular cylindrical thin shells under impact torque. Experiments of impact buckling are made on a Hopkinson torsional bar. The torsional bar gives a step torque on the shells. Through an analysis of the strain-time curve obtained in experiment, the dynamic buckling critical torqueM _er and buckling waves numbern of the shell with different geometric data and some qualitative results are obtained. The buckling behavior of circular cylindrical thin shells under static and impact torque is compared.     

功能梯度夹层多个环形界面裂纹扭转冲击

研究位于功能梯度层和外部均匀材料之间多个环形界面裂纹的扭转冲击问题,功能梯度材料 (FGM)粘结在两种不同的弹性材料之间,功能梯度层和外部材料之间环形界面裂纹的数目是任意的．引进积分变换和位错密度函数将问题化为求解Laplace域里标准的Cauchy奇异积分方程,进而化为求解代数方程;应用Laplace数值反演技术,计算时域里的动应力强度因子(DSIF)．考查了结构几何尺度和材料特性对裂尖动态断裂特性的影响．数值结果表明,DSIF存在一个主峰,到达主峰后,在其相应的静态值附近波动并最终趋于稳定;增加FGM的梯度能减小DSIF的峰值．     

含中心裂纹任意截面柱体扭转时的应力强度因子计算

本文提出了一组应力函数,采用边界配置方法计算了含中心裂纹不同截面形状柱体扭转时的应力强度因子。有关椭圆截面柱体的算例表明,本文方法具有良好的精度。同时,文中给出了圆、椭圆和矩形等不同截面柱体的计算结果。