UNSTEADY THERMAL STRESS ANALYSIS IN AXISYMMETRIC PROBLEMS BY MEANS OF BOUNDARY ELEMENT METHOD

The boundary integral formulation for unsteady thermal stresses in axisymmetric quasi-static problems is proposed. It is shown that axisymmetric unsteady thermal stress problems can be easily solved without the volume integral by means of the thermoelastic displacement potential and boundary element method. It is also shown that the time integral can be easily carried out analytically. In order to investigate the accuracy of rhis method, unsteady thermal stress distributions for a hollow circular cylinder and a torus are obtained.     

柴油机活塞热应力轴对称边界元数值分析

本在推出轴对称热应力问题的边界积分方程的基础上采用２次等参插值函数将边界积分方程离散为边界元方程，进一步运用开发的程序分析了某１３５柴油机活塞的热应力场，分析结果表明边界元法在动力机械零部件热应力问题的分析中，与其它数值方法相比，具有计算精度高，数据准备简单和计算时间少等独特的优点。     

TREATMENT OF THERMOELASTIC EFFECTS IN THE AXISYMMETRIC BOUNDARY INTEGRAL EQUATION METHOD

Unlike any other numerical methods, the axisymmetric form of the Boundary Integral Equation (BIE) method is not a straightforward modification of the two- or three-dimensional formulations. In this paper, axisymmetric thermoelastic effects are treated as effective body forces over the volume and transformed to the boundary to preserve the important advantage of the BIE method of reducing the dimensionality by one. Two alternative approaches are presented; one is based on integrating the already established three-dimensional solutions, and the other assumes from the outset an axisymmetric ring of load rather than the three-dimensional point load. Both approaches are shown to arrive at identical solutions. A numerical example is included to illustrate the numerical implementation and accuracy of the thermoeleastic axisymmetric BIE formulation.     

Analysis of Mode-III fracture problem with multiple cracks by boundary element alternating method

An efficient boundary element alternating method was developed in this present study for the analysis of a Mode-III fracture problem with single or multiple cracks in a finite sheet. Firstly, an analytical solution for a crack in an infinite domain, subjected to an arbitrary longitudinal shear loading across the crack surface, is developed herein. The solution can then be obtained through the iterative superposition of this analytical solution and the boundary element of a finite uncracked sheet. Several Mode-III fracture problems in a finite sheet, with single and multiple cracks under various boundary conditions, are discussed for confirming the validity of this work. Excellent agreement can be observed. The interaction effects among cracks and influence of the boundaries are also considered.     

Stress intensity factors for internal and external cracks in pressurised thick-walled cylinders

Stress intensity factors for both internal and external semi-circular and semi-elliptical surface cracks in internally pressurised thick-walled cylinders of radius ratios between 2 and 3 are presented for a wide range of crack sizes. These solutions were obtained using the boundary integral equation (BIE) method for three-dimensional numerical stress analysis. Hoop strain distributions at the outer circumference of the cylinder are also presented for some external cracks, and shown to be useful for experimentally monitoring crack growth.     

A finite-volume approach to the stress analysis of pressurized axisymmetric structures

A finite-volume procedure for determining the displacement fields and elastic stress distributions within structures that have axisymmetric geometries is presented. The production of a system of linear algebraic equations from the equilibrium equations of the cells that are used to represent the structure is described in detail. An iterative technique is employed to solve these equations and provide the displacement field from which the strain and stress fields can be found. The procedure is used to model two axisymmetric test problems and the stress distribution predicted by the formulation using cell meshes of increasing refinement is compared to analytical solutions and the variations predicted by various finite element procedures.     

IMPORTANCE OF INERTIA TERM IN DYNAMIC CRACK PROBLEMS CONSIDERING LORD–SHULMAN THEORY OF THERMOELASTICITY

A numerical technique is presented for the accurate calculation of stress intensity factors as a function of time for generalized coupled thermoelastic problems. In this task, the effect of the inertia term is investigated, considering different theories of thermoelasticity, and its importance is shown.

A boundary element method using the Laplace transform in time domain is developed for the analysis of fracture mechanics; dynamic coupled thermoelasticity problems with relaxation time are considered in the two-dimensional finite domain. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method. Actual physical quantities in the time domain are obtained using the numerical inversion of the Laplace transform method.

The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. The thermal dynamic stress intensity factor for mode I is evaluated using the J-integral method. The accuracy of the method is investigated through comparison of the results with the data available in literature.

The J integral, which represents the dynamic energy release rate for propagating cracks, contains a boundary integral and a domain integral. The boundary integral contains strain energy, tractions, and strains whereas the domain integral contains inertia and strains. The J-integral method allows these two terms to be calculated separately. In this way, the importance of each term may be investigated by considering different theories of dynamic thermoelasticity.     

Pressure loading and bending of hollow tori

The three-dimensional (3D) Boundary Integral Equation (BIE) method is applied to elastostatic problems involving toroidal geometries. Verification of the accuracy of the 3D method is made by comparing with results from the axisymmetric Finite Element (FE) method and the axisymmetric BIE method. The loading considered for the verification is internal pressure and the geometries include hollow tori of circular and elliptical cross-sections. Results from the BIE method are then presented and compared with results from the Toroidal Elasticity (TE) theory of H. A. Lang. Cases covered are internal pressure loading of a hollow torus and in-plane, out-of-plane and twist bending of a thick-walled 90° elbow. The comparisons are useful in establishing the number of terms required in the TE solutions.     

Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational B-splines

This work explores the application of isogeometric scaled boundary method in the two-dimensional thermoelastic problems of irregular geometry. The proposed method inherits the advantages of both isogeometric analysis and scaled boundary finite element method and overcomes their respective disadvantages. In the proposed approach, the boundaries of the problem domain are discretized with nonuniform rational B-splines (NURBS) basis functions, while the temperature distributions inside the domain are represented by a sequence of power functions in terms of radial coordinate within the framework of scaled boundary finite element method. The resulting solution of the stress in radial direction can be computed analytically for the temperature changes. The construction of tensor product structure is circumvented for the two-dimensional problems as only the boundary information of the problem domain is required. Hence, the flexibility to represent the complex geometry can be significantly improved in the proposed method. Numerical examples are presented to validate the performance of the proposed method where it is seen that superior accuracy, e?ciency, and convergence behavior can be achieved over the conventional scaled boundary finite element method.     

The application of fracture mechanics in thermally stressed structures

There is considerable interest in calculating stress intensity factors at crack tips in thermally stressed structures, particularly in the power generation industry where the safe operation of both conventional and nuclear plant is founded on rigorous safety cases. Analytical methods to study such problems are of limited scope, although they can be extended by introducing numerical techniques. Purpose built numerical methods, however, offer a much greater and more accurate solution capability and in particular the finite element method is well advanced. Such methods are described, including how stress intensity factors can be obtained from the finite element results. They are then applied to a range of thermally stressed problems including plates with central cracks and cylinders with axial and circumferential cracks. Both steady state and transient temperature distributions arising from typical thermal shocks are considered.     

内燃机活塞温度场二次等参轴对称边界元分析

在推导出轴对称势问题的基本解及边界积分方程的基础上,本文采用二次等参插值函数进行离散,形成轴对称热传导问题的边界元方程,编制了包含三类边界条件在内的轴对称边界元分析程序,最后对195柴油机活塞温度场进行了计算。结果表明,边界元法分析活塞温度场,不仅计算准备工作量比有限元大为减少,而且精度高、计算时间短,尤其采用二次等参元后,其优点更加突出。     

Parametric studies of cylindrical pressure vessels with different end closures

Recent developments in the finite element technique using incremental elements permit an easier and more precise determination of stresses, strains and displacements in cylindrical pressure vessels having different end closures. In this paper the geometry analysed is a cylindrical pressure vessel having hemispherical, torispherical, semiellipsoidal and toriconical heads. An axisymmetric solid finite element program employing incremental elements was used for a useful range of vessel parameters.

The formulation of the method used and the results of the parametric study obtained when internal pressure is applied to the vessel are presented. Results are reported in the form of stress intensity parameters based upon the mean circumferential stress of the cylindrical portion of the vessel away from the cylinder head junction.     

The thermal and mechanical behaviour of structural steel piping systems

The temperature, the deformation and the stress field in thermo-mechanical problems play a very important role in engineering applications. This paper presents a finite element algorithm developed to perform the thermal and mechanical analysis of structural steel piping systems subjected to elevated temperatures. The new pipe element with 22 degrees of freedom has a displacement field that results from the superposition of a beam displacement, with the displacement field associated with the section distortion. Having determined the temperature field, the consequent thermal displacement produced in the piping systems due to the thermal variation can be calculated. The temperature rise produces thermal expansion and a consequent increase of pipe length in the structural elements. For small values of the ratio of the pipe thickness to mean radius, the thermal behaviour can be calculated with adequate precision using a one-dimensional mesh approach, with thermal boundary conditions of an axisymmetric type across the pipe section. With this condition, several case studies of piping systems subjected to elevated temperatures and mechanical loads are presented and compared with corresponding results from commercial finite element codes. The main advantage of this formulation is associated with reduced time for mesh generation with a low number of elements and nodes. Considerable computational effort may be saved with the use of this finite pipe element.     

Generalized coupled transient thermoelastic problem of multilayered hollow cylinder with hybrid boundary conditions

The problems of the one-dimensional axisymmetric quasi-static coupled thermoelastic for multilayered hollow cylinder with clamped surface subjected to time-dependent boundary conditions. The formulation begins with the basic equations of thermoelasticity in polar coordinates. The results are obtained employing a method based on a hybrid Laplace transformation matrix similarity transformation and finite difference method. It was shown that the solutions are rapidly convergent. Solutions for the temperature, displacement and thermal stress distributions in both transient and steady state are obtained. The present method can obtain stable solutions at a specific time; thus it is a further concluded that the method and the computing process of the coupled transient thermoelastic problems of multilayered hollow cylinder are powerful and efficient.     

Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks

In this article, a modified state-based peridynamic (PD) method is proposed to solve transient heat conduction problems in functionally gradient materials (FGMs) with extending insulated cracks. The PD formulation of transient heat conduction has been derived by using the time integration through the forward difference technique. Numerical simulations have been performed to verify the accuracy and effectiveness of the proposed method. The analytical solution and the finite element method results are used for comparison. In this work, the material properties of functionally graded materials are assumed to vary exponentially in z-direction. Our PD results show good agreement with analytical solutions and results from the finite element method. Hence the proposed PD method is suitable to deal with the transient heat conduction problem in FGMs with extending insulated cracks.     

Matrix free meshless method for transient heat conduction problems

In the paper, the element free Galerkin method (EFGM) is applied to calculate two-dimensional unsteady state heat conduction problems. As is well known, most of the meshless methods have higher computational cost than that of finite element method (FEM). In order to overcome this shortcoming especially for transient heat conduction problems, mass lumping procedure is adopted in EFGM, which can decrease the computational cost evidently. Moreover, this technique which can simplify the solution procedure makes the essential boundary conditions enforced directly. The results obtained by EFGM combining mass lumping technique are compared with those obtained by finite element method as well as analytical solutions, which shows that the solutions of the present method are in good agreement with FEM’s and analytical solutions.     

ANALYSIS OF MOVING BOUNDARY PROBLEM FOR BUBBLE COLLAPSE IN BINARY SOLUTIONS

Bubble collapse in a binary solution with simultaneous heat and mass transfer is studied under a nonspherically symmetrical condition. A numerical technique is presented in this paper that is suitable for solving the axisymmetric moving boundary problem. Making use of the boundary-fitted curvilinear coordinate system and the Lagrangian method, this finite difference technique is able to dynamically track the evolving bubble shape and size during the collapsing process through adaptive grid regeneration. The grid moving velocity is incorporated as an integral part of the governing equations. The numerical results demonstrate that the combined effect of zero shear stress condition, the decreasing curvature, and short life span results in the absence of a wake behind a collapsing bubble.     

A Laplace transform-based fundamental collocation method for two-dimensional transient heat flow

The fundamental collocation method (FCM) is extended to handle two dimensional transient heat conduction problems in solids. The method is applied in the Laplace transform domain, after which an inversion technique is used to retrieve the time-domain solution. Examples are used to illustrate the method and a technique for evaluating accuracy is discussed. The performance was found to be very good. The method is capable of handling regions of arbitrary shapes, subjected to constant temperature initial conditions and mixed type, time-independent boundary conditions. Due to its inherent advantages over the domain-oriented techniques like the finite element and, finite difference methods, the Laplace transform-based FCM approach presented here may be regarded as a simpler method for solving a wide variety of time-dependent problems in heat conduction and related fields.     

Boundary integral equation technique with application to freezing around a buried pipe

In this paper the use of the boundary integral equation method (BIEM) for multidimensional problems with a moving phase-change interface is explored. The method is shown to be suited to heat transfer problems where the field equations are linear in each region and the boundary or interface matching conditions are both highly irregular and non-linear. For moving interface problems the BIE technique both reduces the dimensions of the problem by one, thus decreasing storage requirements, and directly solves for the unknown normal temperature gradient on each side of the interface for the determination of the instantaneous interface velocity. To illustrate the versatility of this technique the BIEM is applied to a previously unsolved problem: the melting/freezing around a pipe buried in a semi-infinite domain where the melting/freezing is initiated at the free surface and the medium is initially not at the phase-change temperature. For simplicity quasi-steady heat conduction is assumed in both the thawed and frozen zones. Solutions are presented for various values of the governing parameters.     

Elastoplastic crack analysis of thick-walled cylinders using the symmetric-iterative scheme of coupled BE and FE discretizations

This paper uses the symmetric-iterative method of coupled FE and BE discretizations for the solution of elastoplastic thick-walled cylinders with symmetric cracks which are discretized partly by finite elements and partly by boundary elements. The calculations for J-integrals, by means of the present method and the finite element method, agree well.