The torsion problem of a multilayered finite cylinder with the multiple interface cylindrical cracks

The torsion axisymmetric problem for a finite cylinder consisting of an arbitrary quantity of cylindrical coaxial layers is solved. Multiple cylindrical cracks with free of loading branches are situated on adjoining surfaces of the layers. The boundary problem is reduced to the system of integro-differential equations, its solution is found with the help of the orthogonal polynomials method. The novelty of the paper is in the construction of a solution for an arbitrary number of cylinder layers which allows the approximation of the initial problem for functionally graded materials by the problem for coaxial cylinders with jumplike changing elastic constants of the materials. Since the solution is built regardless of the number of layers (the elastic parameters of all layers are included in the constructed solution), one can refine an initial problem’s statement by increasing the number of layers. The stress intensity factors are found for an arbitrary number of cylindrical interface cracks in the multilayered cylinder of a finite length.     

Plastic zone correction in a stretched cylinder with an external circumferential crack

The Dugdale hypothesis is adapted to the problem of an external circumferential crack in a stretched cylinder. The lateral surface of the cylinder is stress free and restrained from radial displacements. An external circumferential edge crack in the cylinder which is considered elastic-perfectly plastic is envisaged with the assumption that the plastic zone forms a very thin in-plane layer surrounding the crack. The solution of the problem is reduced to the solution of dual Dini series which, in turn, is reduced to a Fredholm integral equation of the second kind. Solving this integral equation numerically and using the boundedness of the axial stress, the size of the plastic zone correction is obtained.     

Instability of an Oscillating Cylinder in a Circulation Flow of Ideal Fluid

The problem of the stability of a circular cylinder in a circulation flow is considered under the condition that the cylinder can perform both free (free cylinder) and forced oscillations (cylinder on a spring). It is shown that this simple system can be unstable in the presence of flow vorticity. Particular cases of vorticity distributions which make it possible to obtain an analytic solution are considered. The case of weak monotonically decreasing vorticity of an arbitrary form is analyzed for an arbitrary relation between the densities of the cylinder and the fluid. It turns out that the instability can develop only for a cylinder whose density is greater than that of the fluid. An approximate method of solving this problem based on consideration of the energy balance in the system is constructed. This makes it possible to obtain an expression for the growth rates and explain the physical mechanism realizing the instability, which is associated with the possibility of energy transfer from perturbations in the critical layer to the cylinder oscillations.     

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     

Cavity expansion of a gradient-dependent solid cylinder

This paper presents an elasto-plastic analysis for cavity expansion in a solid cylinder. The solid is modelled using a strain gradient plasticity model to account for the influence of microstructures on the macroscopic mechanical behaviour. A numerical shooting method, together with Broyden’s iteration procedure, is developed to solve the resulting fourth-order ordinary differential equation with two-point boundary conditions for the gradient-dependent problem. Fully elastic-plastic solutions to the cavity expansion are obtained and they are compared with conventional results for a number of examples. The effects of microstructure on macroscopic behaviour for the cavity expansion problem are analysed. It is demonstrated that, with consideration of microstructural effects, the deformation and stress distributions in the cylinder are highly inhomogeneous during both the initial loading and the subsequent elastic and plastic expansion stages. The gradient effects can result in a stiffer response in the elastic regime (as compared with the corresponding conventional prediction), but a weaker response in the plastic regime. As expected, the overall elasto-plastic behaviour of the gradient-dependent cylinder depends on the material parameters as well as the cylinder thickness. It is shown that the strain gradient theory solutions reduce to the conventional ones as a special case when the dimension of the microstructures is negligible compared with the cylinder size. The results in this paper can be used as a benchmark for further numerical investigations of the cavity expansion problem.     

Transient thermal elastic fracture of a piezoelectric cylinder specimen

A finite piezoelectric cylinder with an embedded penny-shaped crack is investigated for a thermal shock load on the outer surface of the cylinder. The theory of linear electro-elasticity is applied to solve the transient temperature field and the associated thermal stresses and electrical displacements without crack. These thermal stresses and electrical displacements are added to the surfaces of the crack to form an electromechanical coupling and mixed mode boundary-value problem. The electrically permeable crack face boundary condition assumption is used, and the thermal stress intensity factor and electrical displacement intensity factor at the crack border are evaluated. The thermal shock resistance of the piezoelectric cylinder is evaluated for the analysis of piezoelectric material failure in practical engineering applications.     

3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid

The free vibration of an arbitrarily thick orthotropic piezoelectric hollow cylinder with a functionally graded property along the thickness direction and filled with a non-viscous compressible fluid medium is investigated. The analysis is directly based on the three-dimensional exact equations of piezoelasticity using the so-called state space formulations. The original functionally graded shell is approximated by a laminate model, of which the solution will gradually approach the exact one when the number of layers increases. The effect of internal fluid can be taken into consideration by imposing a relation between the fluid pressure and the radial displacement at the interface. Analytical frequency equations are derived for different electrical boundary conditions at two cylindrical surfaces. As particular cases, free vibration of multi-layered piezoelectric hollow cylinder and wave propagation in infinite homogeneous cylinder are studied. Numerical comparison with available results is made and dispersion curves predicted from the present three-dimensional analysis are given. Numerical examples are further performed to investigate the effects of various parameters on the natural frequencies.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

A multi-scale constitutive model for the sintering of an air-plasma-sprayed thermal barrier coating, and its response under hot isostatic pressing

A micromechanical model is developed for the sintering of an air-plasma-sprayed, thermal barrier coating, and is used to make predictions of microstructure evolution under free sintering and under hot isostatic pressing. It is assumed that the splats of the coating are separated by penny-shaped cracks; the faces of these cracks progressively sinter together at contacting asperities, initially by the mechanism of plastic yield and subsequently by interfacial diffusion. Diffusion is driven by the reduction in interfacial energy at the developing contacts of the cracks and also by the local contact stress at asperities. The contact stress arises from the remote applied stress and from mechanical wedging of the rough crack surfaces. Sintering of the cracks leads to an elevation in both the macroscopic Young's modulus and thermal conductivity of the coating, and thereby leads to a degradation in thermal performance and durability. An assessment is made of the relative roles of surface energy, applied stress and crack face roughness upon the sintering response and upon the evolution of the pertinent mechanical and physical properties. The evolution in microstructure is predicted for free sintering and for hot isostatic pressing in order to provide guidance for experimental validation of the micromechanical model.     

Initial stage of the circular cylindermotion in a fluid after an impact with the formation of a cavity

The joint motion of an ideal fluid and a submerged circular cylinder is considered in the initial stage after an impact. The dynamics of separation points on the inner free boundary (cavity boundary) and the shapes of the inner and outer free boundaries of the fluid are determined. An asymptotic analysis of the inner free boundary near the separation points is made. The effects of the Froude number, the pressure difference, and the cylinder immersion depth are investigated.     

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.     

Instability of Bingham fluids in Taylor-Dean flow between two concentric cylinders at arbitrary gap spacings

Stability of Bingham fluids is investigated numerically in azimuthal pressure-driven flow between two infinitely long concentric cylinders. An infinitesimal perturbation is introduced to the basic flow and its time evolution is monitored using normal mode linear stability analysis. An eigenvalue problem is obtained which is solved numerically using pseudo-spectral collocation method. Numerical results are obtained for two different cases: (i) the inner cylinder is rotating at constant velocity while the outer cylinder is fixed (i.e., the Taylor-Dean flow) and (ii) both cylinders are fixed (i.e., the Dean flow). The results show that the yield stress always has a stabilizing effect on the Taylor-Dean flow. But, for the Dean flow the effect of the yield stress is predicted to be stabilizing or destabilizing depending on the magnitude of the Bingham number and also the gap size.     

An interface crack with partially electrically conductive crack faces under antiplane mechanical and in-plane electric loadings

An interface crack in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the crack faces is electrically conductive while the other part is electrically permeable. All electro-mechanical values are presented using sectionally-analytic vector-functions and a combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Simple analytical expressions for the shear stress, electric field and also for mechanical displacement jump of the crack faces are derived. These values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear stress, electric field and electric displacement jump are found. Their intensity factors are determined as well. Intensity factors variations with respect to the external electric field and different ratios between the electrically conductive and electrically permeable crack face zones are also demonstrated.     

Some Exactly Solvable Problems of the Radiation of Three–Dimensional Periodic Internal Waves

An exact solution of the problem of the generation of three–dimensional periodic internal waves in an exponentially stratified, viscous fluid is constructed in a linear approximation. The wave source is an arbitrary part of the surface of a vertical circular cylinder which moves in radial, azimuthal, and vertical directions. Solutions satisfying exact boundary conditions, describe both the beam of outgoing waves and wave boundary layers of two types: internal boundary layers, whose thickness depends on the buoyancy frequency and the geometry of the problem, and viscous boundary layers, which, as in a homogeneous fluid, are determined by kinematic viscosity and frequency. Asymptotic solutions are derived in explicit form for cylinders of large, intermediate, and small dimensions relative to the natural scales of the problem.     

轴对称横观各向同性热弹性圆柱的分解

为了得到精确的应力场、位移场、温度场,将扭转圆轴的精化理论研究方法推广到轴对称横观各向同性热弹性圆柱。利用Bessel函数以及轴对称横观各向同性热弹性圆柱的通解,给出了轴对称横观各向同性热弹性圆柱的分解定理。根据柱面齐次边界条件获得了精确的精化方程,精化方程可以分解为一阶方程、超越方程、温度方程,从而将横观各向同性热弹性圆柱的轴对称问题分解为轴向拉压问题、超越问题、热-应力耦合问题。超越部分对应端部自平衡情况,可以清晰地了解到端部应力分布对内部应力场的影响,热-应力耦合部分对应无外加应力场时圆柱内部因温度变化引起的热应力。     

A Solution for the Stress Distribution in a Granular Medium

We propose an example of an incremental elastic problem for a granular material. The mechanical behavior of this material is sensitive to the confining pressure that typically is applied before any loading. Experiments, numerical simulations, and theoretical models show that the effective elastic moduli of a granular medium are function of the confining pressure in non-linear way. Therefore, if we consider a reference state where the pressure is not constant the material behaves differently in each point and, for example, the stress associated to a subsequently loading should be obtained as solution of a non-homogeneous material. We focus our attention on this kind of problem for a granular material that fills an hollow cylinder first isotropically compressed and then sheared under a rotatory motion. Next a small perturbation is applied on the boundaries of the specimen and we evaluate the corresponding stress distribution in the plane perpendicular to the axis of the cylinder.     

The effect of the couple stress parameter and Prandtl number on the transient natural convection flow over a vertical cylinder

An analysis is performed to study transient free convective boundary layer flow of a couple stress fluid over a vertical cylinder, in the absence of body couples. The solution of the time-dependent non-linear and coupled governing equations is carried out with the aid of an unconditionally stable Crank-Nicolson type of numerical scheme. Numerical results for the steady-state velocity, temperature as well as the time histories of the skin-friction coefficient and Nus- selt number are presented graphically and discussed. It is seen that for all flow variables as the couple stress control parameter, Co, is amplified, the time required for reaching the temporal maximum increases but the steady-state decreases.     

Free convection of a dusty‐gas flow along a semi‐infinite vertical cylinder

An analysis is performed to study the free convection of a dusty‐gas flow along a semi‐infinite isothermal vertical cylinder. The governing equations of the flow problem are transformed into non‐dimensional form and the resulting nonlinear, coupled parabolic partial differential equations have been solved numerically using an implicit finite difference scheme of Crank–Nicholson type. The flow variables such as gas–velocity, dust‐particle velocity and temperature, shearing stress and heat transfer coefficients are calculated numerically for various parameters occurring in the problem. It is observed that due to the presence of dust particles, the gas velocity is found to decrease. Copyright © 2009 John Wiley & Sons, Ltd.