Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lamé problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.     

Modeling of progressive delamination in a thin film driven by diffusion-induced stresses

A semi-analytical method based on the cohesive model has been developed to investigate the progressive growth of interface delamination in an axisymmetric thin film electrode driven by diffusion-induced stresses under the assumption that the electrode remains elastic during the Li-ion diffusion process. The evolutions of the cohesive zone and debonding zone with respect to charging time have been predicted. The cohesive zone propagates in an accelerating manner and the debonding zone advances in a slowing down manner. The key parameters that control the interfacial stresses and delamination have been identified from the obtained governing equations. And according to the discussions on the key parameters, design insights into the geometry, charging velocity and material properties of the electrode have been provided.     

Practical Green's function for the thermal stress field induced by a heat source in plane thermoelasticity

The classical Green’s functions used in the literature for a heat source in a homogeneous elastic medium cannot lead to finite remote thermal stresses in the medium, so that they may not work well in practical thermal stress analyses. In this paper, we develop a practical Green’s function for a heat source disposed eccentrically into an elastic disk/cylinder subject to plane deformation. The edge of the disk/cylinder is assumed to be thermally permeable and traction-free. The full thermal stress field induced by the heat source in the disk/cylinder is determined exactly and explicitly via the Cauchy integral techniques. In particular, a very simple formula is obtained to describe the hoop thermal stress on the edge of the disk/cylinder, which may be conveniently useful for analyzing the thermal stresses in microelectronic components.     

Temperature and stress analysis of an elastic circular cylinder in contact with heated rigid stamps

Summary  The present paper discusses a plane strain problem of transient thermoelasticity in a circular cylinder which is in partial contact with two heated rigid stamps, in the case where the coefficient of relative heat transfer on the contact surface of the cylinder is different from that on the traction-free surface. A finite difference method with respect to the time variable and Airy's thermal stress function is employed to analyze the temperature and thermoelastic fields. The problem is formulated in terms of two dual-series equations derived not only from the thermal boundary conditions but also from the mechanical boundary conditions. Since the radial, hoop and axial stresses have singularities at the end of the contact surface of the cylinder, the stress singularity coefficients are defined and then the relationship among these three coefficients is also obtained. Finally, numerical results are illustrated graphically. Received 3 March 2000; accepted for publication 12 July 2000     

Analytical modeling and simulation of porous electrodes: Li-ion distribution and diffusion-induced stress

A new model of porous electrodes based on the Gibbs free energy is developed, in which lithium-ion(Liion) diffusion, diffusion-induced stress(DIS), Butler–Volmer(BV) reaction kinetics, and size polydispersity of electrode particles are considered. The influence of BV reaction kinetics and concentration-dependent exchange current density(ECD) on concentration profile and DIS evolution are numerically investigated. BV reaction kinetics leads to a decrease in Li-ion concentration and DIS. In addition, concentrationdependent ECD results in a decrease in Li-ion concentration and an increase in DIS. Size polydispersity of electrode particles significantly affects the concentration profile and DIS.Optimal macroscopic state of charge(SOC) should consider the influence of the microscopic SOC values and mass fractions of differently sized particles.     

A note on critical reflections of axially symmetric waves in an isotropic,elastic, hollow cylinder

At critical incidence of equivoluminal waves, the wave motion is evanescent in the case of an isotropic, elastic, hollow cylinder with traction-free cylindrical surfaces. However, it is shown that in this critical situation the wave motion can be obtained by using a suitable limiting procedure.     

Analysis of anisotropic sector with a radial crack under anti-plane shear loading

In this paper, the anti-plane shear deformation of an anisotropic sector with a radial crack is investigated. The traction–traction boundary conditions are imposed on the radial edges and the traction-free condition is considered on the circular segment of the sector. A novel mathematical technique is employed for the solution of the problem. This technique consists of the use of some recently proposed finite complex transforms (Shahani, 1999), which have complex analogies to the standard finite Mellin transforms of the first and second kinds. However, it is essential to state the traction-free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method. The resultant dual integral equations are solved numerically to determine the stress intensity factors at the crack tips. In the special cases, the obtained results coincide with those cited in the literature.     

Exact solutions for radial deformations of a functionally graded isotropic and incompressible second-order elastic cylinder

We find closed-form solutions for axisymmetric plane strain deformations of a functionally graded circular cylinder comprised of an isotropic and incompressible second-order elastic material with moduli varying only in the radial direction. Cylinder's inner and outer surfaces are loaded by hydrostatic pressures. These solutions are specialized to cases where only one of the two surfaces is loaded. It is found that for a linear through-the-thickness variation of the elastic moduli, the hoop stress for the first-order solution (or in a cylinder comprised of a linear elastic material) is a constant but that for the second-order solution varies through the thickness. The radial displacement, the radial stress and the hoop stress do not depend upon the second-order elastic constant but the hydrostatic pressure and hence the axial stress depends upon it. When the two elastic moduli vary as the radius raised to the power two or four, the radial and the hoop stresses in an infinite space with a pressurized cylindrical cavity equal the pressure in the cavity. For an affine variation of the elastic moduli, the hoop stress in an internally loaded cylinder made of a linear elastic isotropic and incompressible material at the point is the same as that in a homogeneous cylinder. Here R_in and R_ou equal, respectively, the inner and the outer radius of the undeformed cylinder and R the radial coordinate of a point in the unstressed reference configuration.     

A simple model for diffusion-induced dislocations during the lithiation of crystalline materials

Assuming that the lithiation reaction occurs randomly in individual small particles in the vicinity of the reaction front, a simple model of diffusion-induced dislocations was developed. The diffusion-induced dislocations are controlled by the misfit strain created by the diffusion of solute atoms or the phase transformation in the vicinity of the reaction front. The dislocation density is proportional to the total surface area of the “lithiated particle” and inversely proportional to the particle volume. The diffusion-induced dislocations relieve the diffusion-induced stresses.     

Determination of the thermal stress wave propagation in orthotropic hollow cylinder based on classical theory of thermoelasticity

The thermoelasticity problem in a thick-walled orthotropic hollow cylinder is solved analytically using finite Hankel transform and Laplace transform. Time-dependent thermal and mechanical boundary conditions are applied on the inner and the outer surfaces of the cylinder. For solving the energy equation, the temperature itself is considered as boundary condition to be applied on both the inner and the outer surfaces of the orthotropic cylinder. Two different cases are assumed for solving the equation of motion: traction–traction problem (tractions are prescribed on both the inner and the outer surfaces) and traction–displacement (traction is prescribed on the inner surface and displacement is prescribed on the outer surface of the hollow orthotropic cylinder). Due to considering uncoupled theory, after obtaining temperature distribution, the dynamical structural problem is solved and closed-form relations are derived for radial displacement, radial and hoop stress. As a case study, exponentially decaying temperature with respect to time is prescribed on the inner surface of the cylinder and the temperature of the outer surface is considered to be zero. Owing to solving dynamical problem, the stress wave propagation and its reflections were observed after plotting the results in both cases.     

压电空心圆柱中波的传播

应用三维压电弹性体轴对称模型对压电空心圆柱中波的传播进行了研究. 发现在 圆柱中是否具有压电性质会对波的传播带来显著的差异. 当波长趋向于零时，在压电圆柱中 拟P波的波速渐进趋向于横观各向同性弹性体的准P波波速，而非压电圆柱中拟P波的波速 渐进趋向于一维杆模型中的P波波速；在压电圆柱中拟SV波存在驻波现象. 圆柱中的SH波 同电场无关，所以在压电圆柱和非压电圆柱中SH波具有相同的频散曲线. 应用积分变 换方法将圆柱的控制方程同其侧边界条件相结合，得到了一组动力学方程. 针对具体的侧边 界条件，得到相对应的波导条件和频散方程. 在此基础上通过数值计算模拟了圆柱受到端部 应力脉冲激励后的瞬态响应. 同时讨论了空心圆柱的半径比对轴向波传播的影响.     

Nonlinear diffusion-induced stresses in a long bar of square cross section

This paper is to investigate the nonlinear effect of the self-induced electric field on the diffusion-induced stresses in a long bar. We first approximate the nonlinear concentration-dependent diffusivity as a series of third-degree polynomials by the least-squares curve-fitting techniques, and then calculate the distributions of concentration by the Galerkin method. Afterwards, the diffusion-induced stresses inside the bar are determined analytically by introducing the Goodier displacement potential and Airy stress function. It is found that the nonlinear self-induced electric fields can depress both the concentration gradient and the maximum diffusion-induced stresses apparently, and these effects are more significant at short times than at long times.     

具有表面效应的球形纳米电极颗粒中的反应-扩散-力学全耦合分析

表面效应在纳米电极颗粒中占有主导地位,论文首先建立了锂离子电池充放电过程中考虑表面效应的反应-扩散-力学全耦合模型;然后对比了有无表面效应对锂离子浓度、径向应力和环向应力的影响;最后探讨了反应系数和尺寸效应对浓度和扩散诱导应力的影响.数值结果表明表面效应随着充电时间的增加逐渐减小直至充电结束趋于稳定;表面压效应对本模型的浓度无影响但抑制了扩散诱导应力的增加;较慢的化学反应和较小的纳米电极颗粒尺寸可抑制电极应力的增加.     

Some problems of linear elasticity for cylinders in micropolar orthotropic material

Some special problems for axisymmetric solids made of linearly elastic orthotropic micropolar material with central symmetry are dealt with. The first one is a hollow circular cylinder of unlimited length, subjected to internal and external uniform pressure. The second one is a hollow or solid circular cylinder of finite length, subjected to a relative rotation of the bases about its axis. In both cases, one of the axes of elastic symmetry is parallel to the cylinder axis; the other two are arbitrarily oriented in the plane of any cross-section of the solid. The elastic properties are invariant along the cylinder axis. It is shown that the two problems are governed by formally similar sets of ordinary differential equations in the kinematic fields (in-plane displacements and microrotations). In the general case, numerical solutions are derived. The solution for the cylinder subjected to radial pressure does not significantly differ from that obtained in classical elasticity, at least in terms of radial and hoop force stresses. In the case of a cylinder subjected to torsion the difference between the micropolar and the classical solutions is more pronounced. The torque induces twisting couple stresses about the cylinder axis of variable sign. Finally, size effects in terms of torsional inertia are pointed out.     

A Pressurized Functionally Graded Hollow Cylinder with Arbitrarily Varying Material Properties

The elastic analysis of a pressurized functionally graded material (FGM) annulus or tube is made in this paper. Different from existing studies, this study deals with an axisymmetrical FGM hollow cylinder or disk with arbitrarily varying material properties. A simple and efficient approach is suggested, which reduces the associated problem to solving a Fredholm integral equation. The resulting equation is approximately solved by expanding the solution as series of Legendre polynomials. The stresses and displacements can be represented in terms of the solution to the equation. For radius-dependent Young’s modulus, numerical results of the distribution of the radial and circumferential stresses are presented graphically. Our results indicate that change in the gradient of the FGM tube does not produce a substantial variation of the radial stress, but strongly affects the distribution of the hoop stress. In particular, the hoop stress may reach its maximum at an internal position or at the outer surface when the tube is internally pressurized. The results obtained are helpful in designing FGM cylindrical vessels to prevent failure.      

Inflation and eversion of functionally graded non-linear elastic incompressible circular cylinders

We study axisymmetric radial deformations of a circular cylinder composed of an inhomogeneous Mooney-Rivlin material with the two material parameters varying continuously through the cylinder thickness either by a power law or an affine relation. It is found that for the exponent of the power law function equal to 1, the hoop stress for an internally pressurized cylinder is uniform in the cylinder. One can tailor the gradation of these two material parameters to make the maximum tensile hoop stress occur either on the inner surface or on the outer surface. Also, the stress concentration in a pressurized thick cylinder strongly depends upon the value of the exponent of the power law variation of the two material parameters. For an affine through-the-thickness variation of the two elastic moduli the hoop stress at the point is nearly the same as that in a cylinder composed of a homogeneous material. Here R_in and R_ou equal, respectively, the inner and the outer radii of the cylinder in the unstressed reference configuration, and R is the radial coordinate of a point in the reference configuration. The stress distribution in an everted cylinder strongly depends upon its thickness in the reference configuration.     

The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linearly Elastic Materials

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic hollow circular cylinders or disks under uniform internal or external pressure. 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 analog of the classic Lamé problem for a pressurized homogeneous isotropic hollow circular cylinder or disk is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. It is shown that the stress response of the inhomogeneous cylinder (or disk) is significantly different from that of the homogeneous body. For example, the maximum hoop stress does not, in general, occur on the inner surface in contrast with the situation for the homogeneous material. The results are illustrated using a specific radially inhomogeneous material model for which explicit exact solutions are obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Experimental evidence of diffusion-induced bias in near-wall velocimetry using quantum dot measurements

Near-surface velocity measurements are carried out with quantum dot (QD) nanoparticles using evanescent wave illumination. Relying on the small size of QDs, their correspondingly small hydrodynamic radius and high Brownian diffusion coefficient, we consider the situation where the tracer diffusion length over the inter-frame time Δt is large compared to the size of the interrogation region next to the wall. While keeping all other experimental parameters fixed, we systematically increase Δt by as much as a factor of 25, resulting in an increase of the QD diffusion length by a factor of 5. Data indicate a significant overestimation of the “apparent” mean velocity measured experimentally. These results provide a direct confirmation of the phenomenon of diffusion-induced bias described by the simulations of Sadr et al. (2007).     

Elastic analyses of heterogeneous hollow cylinders

Two different kinds of heterogeneous elastic hollow cylinders are studied in the present paper. One is a multi-layered cylinder with different values in different layers for both elastic modulus and Poisson’s ratio. Another is an elastic hollow cylinder with continuously graded material properties. By introducing two recursive algorithms, the extrusion stresses between two neighbor layers in the multi-layered cylinder submitted to uniform pressures on the inner and outer surfaces can be simply determined. Then the exact solutions of the multi-layered structure can be found based on Lamé’s solution. For the hollow cylinder with continuously graded properties, the displacement method is used. Both Whittaker equation and hyper-geometric equation are derived and successfully solved, and then the exact solutions are found. The results obtained in the present paper are compared with the numerical solutions and good agreements are found. At the end of the present paper, some inherent properties of these two different kinds of heterogeneous elastic hollow cylinders are presented and discussed. The results obtained in the present paper are useful in the design and analysis for composites reinforced by unidirectional fiber layers.     

Surface tension-induced stress concentration around an elliptical hole in an anisotropic half-plane

We examine the surface tension-induced stress concentration around an elliptical hole inside an anisotropic half-plane with traction-free surface. Using conformal mapping techniques, the corresponding complex potential in the half-plane is expressed in a series whose unknown coefficients are determined numerically. Our results indicate that the maximum hoop stress around the hole (which appears in the vicinity of the point of maximum curvature) increases rapidly with decreasing distance between the hole and the free surface. In particular, for an elliptical or even circular hole in an anisotropic half-plane we find that, with decreasing distance between the hole and the free surface, the hoop stress can switch from compressive to tensile at certain points on the hole's boundary and from tensile to compressive at others. This phenomenon is absent in the case of an elliptical or even circular hole in the corresponding case of an isotropic half-plane.