Axisymmetric Elasticity Problems Research Articles

Application of the Least Squares Method in Axisymmetric Biharmonic Problems

An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.

A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder

A variational method of homogeneous solutions for axisymmetric elasticity problems for cylinder

High-accuracy quadrature methods for solving boundary integral equations of axisymmetric elasticity problems

We investigate the mechanical quadrature methods (MQMs) for solving boundary integral equations (BIEs) of the three-dimensional axisymmetric elasticity equations with Dirichlet’s problems. The convergence of the MQMs is proved based on Anselone’s collective compact theory. Moreover, an asymptotic error expansion of the MQMs shows that the approximation order is of O(h3). Numerical examples are tested and the results verify the theoretical analysis.

Exact Solution of Some Axisymmetric Problems for Elastic Cylinders of Finite Length Taking Into Account Specific Weight

The finite Hankel transform is used to find the exact solution of two axisymmetric problems of elasticity for solid and hollow cylinders of finite length taking their specific weight into account. The boundary conditions at the lower end and on the lateral faces of the cylinders are sliding restraint. Axisymmetric normal and tangential forces act on the upper end. The transforms of the displacements and stresses are obtained. The calculated normalized stresses on the outside and inside cylindrical surfaces are plotted for different height ratios

Analytical solutions to the axisymmetric elasticity and thermoelasticity problems for an arbitrarily inhomogeneous layer

In this paper, we present a technique for constructing an analytical solution to the axisymmetric elasticity and thermoelasticity problems in terms of stresses for an inhomogeneous layer, whose elastic and thermophysical properties vary arbitrarily within the thickness-coordinate. By making use of the direct integration method, the equilibrium and compatibility equations are reduced to the governing Volterra-type integral equations accompanied with both integral and local boundary conditions for the key functions. To solve the obtained governing equations, we employed the resolvent-kernel technique which results in closed-form analytical expressions for the key functions. Having determined the key functions, the stress-tensor components are found through the relationship established by the integration of equilibrium equations. The same solution procedure is employed for solving the steady-state heat-conduction problem in an inhomogeneous layer. Typical numerical examples are discussed.

Determination of hydrofracture geometry in a production reservoir

The article describes numerical modeling of growth of five successive hydraulic fractures in the directions transverse to the well, in plane and axisymmetric elastic problems, under assumption of perfect breakdown fluid. The zone of influence of the existing created fractures on the newly initiating hydrofractures is defined, and the pressure for a new fracture to grow is evaluated. Parameters governing distortion of fractures are determined. The calculations in the framework of the plane and axisymmetric problems are compared.

Variational Method of Homogeneous Solutions in Axisymmetric Elasticity Problems for a Semiinfinite Cylinder

UDC 539.3 We develop a variational method of homogeneous solutions for the solution of axisymmetric elasticity problems for a semiinfinite cylinder with free lateral surface. We consider four types of boundary conditions imposed on the end of cylinder, namely, the conditions in stresses, in displacements, and two types of mixed conditions. The solution of the problems with the help of this method is reduced to the solution of infinite systems of linear algebraic equations. We perform the numerical analysis of convergence of the obtained solutions. We also consider an example of application of the proposed approach to the determination of stress concentration near the joint of the end of the cylinder with a perfectly rigid lateral surface. The problems of determination of the stress-strain state of cylindrical bodies appear in various scientific disciplines and, in particular, in the applied theories of the strength of solids [9]. Despite significant advances in the development of the methods aimed at the numerical analysis of the problems of mechanics and a wide choice of certified highly efficient software environments intended for the solution of these problems with the help of

Equilibrium of an elastic finite cylinder under axisymmetric discontinuous normal loadings

This article presents an analytical technique for solving the axisymmetric elasticity problem for a finite solid cylinder subjected to discontinuous normal loadings on its ends and lateral surface. This technique is based on application of the method of crosswise superposition by representing the solution for stresses in the form of decompositions into Fourier and Bessel–Dini series. For determination of the coefficients in these series, the infinite systems of linear algebraic equations are obtained and solved by means of a modified algorithm of advanced reduction. The technique is numerically validated for typical cases of discontinuous loading. It is shown that the solution procedure is efficient for determination of the stresses in the cylinder including its edges and discontinuity points of normal loadings.

Transition finite element families for adaptive analysis of axisymmetric elasticity problems

In this paper, four transition element families that comprise five- to seven-node quadrilateral elements are developed based on the hybrid-stress and enhanced assumed strain (EAS) formulations for adaptive analyses of axisymmetric elasticity problems. For members in the first hybrid-stress family, a quasi-linear stress field with ten equilibrating stress modes is derived and employed. To study the effect of including more stress modes in the assumed stress field, another family with two additional stress modes is implemented. On the other hand, two EAS element families are constructed with respect to the incompatible displacement modes of two existing incompatible displacement transition element families. Several numerical examples are exercised. It can be seen that the first hybrid-stress family is the most accurate one among the proposed families. Moreover, the EAS families are close to the respective incompatible families in accuracy yet the former families are not only more efficient in computation but also more concise in formulation.

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D- n model can all be explicitly recovered by the current general solution.

Representations of elastic fields of circular dislocation and disclination loops in terms of spherical harmonics and their application to various problems of the theory of defects

Elastic fields of circular dislocation and disclination loops are represented in explicit form in terms of spherical harmonics, i.e. via series with Legendre and associated Legendre polynomials. Representations are obtained by expanding Lipschitz-Hankel integrals with two Bessel functions into Legendre series. Found representations are then applied to the solutions of elasticity boundary-value problems of the theory of defects and to the calculation of elastic fields of segmented spherical inclusions. In the framework of virtual circular dislocation–disclination loops technique, a general scheme to solving axisymmetric elasticity problems with boundary conditions specified on a sphere is given. New solutions for elastic fields of a twist disclination loop in a spherical particle and near a spherical pore are demonstrated. The easy and straightforward way for calculations of elastic fields of segmented spherical inclusion with uniaxial eigenstrain is shown.

J-integral calculation by domain integral technique using adaptive finite element method

An adaptive finite element method for analyzing two-dimensional and axisymmetric nonlinear elastic fracture mechanics problems with cracks is presented. The J-integral is used as a parameter to characterize the severity of stresses and deformation near crack tips. The domain integral technique, for which all relevant quantities are integrated over any arbitrary element areas around the crack tips, is utilized as the J-integral solution scheme with 9-node degenerated crack tip elements. The solution accuracy is further improved by incorporating an error estimation procedure onto a remeshing algorithm with a solution mapping scheme to resume the analysis at a particular load level after the adaptive remeshing technique has been applied. Several benchmark problems are analyzed to evaluate the efficiency of the combined domain integral technique and the adaptive finite element method.

10402 球状弾性介在物をもつ弾性円柱の引張り(材料(1))

This paper deals with a stress concentration problem of an elastic circular solid cylinder with an elastic spherical inclusion under tension. Effects of Poisson's ratio, especially effects of negative Poisson's ratio, on the stress concentrations are investigated. Lakes has shown that there are materials with negative Poisson's ratios. A method of solution is presented for the problem by applying Green's functions for axisymmetric body force problems of infinite circular solid cylinder and fundamental solutions for axisymmetric problems of elasticity.

Numerical solution of axisymmetric nonlinear elastic problems including shells using the theory of a Cosserat point

Starting with a recently developed three-dimensional eight node brick Cosserat element for nonlinear elastic materials, a simplified Cosserat element is developed for torsionless axisymmetric motions. The equations are developed within the context of the theory of a Cosserat point and the resulting theory is hyperelastic and is valid for dynamics of nonlinear elastic materials. The axisymmetric Cosserat element has four nodes with a total of eight degrees of freedom. As in the more general element, the constitutive equations are algebraic expressions determined by derivatives of a strain energy function and no integration is needed throughout the element region. Examples of large deformations of a nearly incompressible circular cylindrical tube and large deflections of a compressible clamped circular plate are considered to test the accuracy of the element.

Weak formulation of axi-symmetric frictionless contact problems with boundary elements: Application to interface cracks

A boundary element formulation of the axi-symmetric elastic problem, including contact conditions for multi-domain problems, is presented in this paper and applied to the analysis of interface cracks. A weak formulation of the equilibrium and compatibility equations has been developed to allow non-conforming discretizations to be used in the contacting boundaries. In order to obtain a high accuracy in the solution (employing continuous linear elements), the singular terms appearing in the axi-symmetric boundary integral equations have been integrated analytically. The numerical analysis of a penny-shaped crack at the interface of two dissimilar materials and of a fibre–matrix debonding crack in the single fibre fragmentation test is presented.

A Boussinesq's Problem of an Elastic Half Space with an Elastic Spherical Inclusion

The problem of determing of stresses and displacements of an elastic half space subjected to a concentrated force acting at a point of the plane surface is called Boussinesq's problem. This paper deals with Boussinesq's problems of an elastic half space with an elastic spherical inclusion. We use a body force distribution method for the stress concentration problem. For this purpose, we apply the Green's functions for axisymmetric body force problems of an elastic half space and the fundamental solution for axisymmetric body force problems of elasticity. The problem is formulated by an integral equation with unknown functions corresponding to the intensities of distributed body forces. Stress distributions around the inclusion are shown by numerical calculations and influences of elastic moduli on the stresses are also investigated.

1401 2個の球体剛体介在物をもつ弾性体の熱応力(S18-1 介在物・空孔・き裂,S18 不規則性,不均一性構造・材料の数理モデルとその応用)

This paper deals with the thermal stresses of an infinite elastic solid with two rigid spherical inclusions under a uniform temperature field. A body force distribution method of solution is presented for the problem by applying Green's functions for axisymmetric problems of elasticity. The problem is formulated by an integral equation with unknown body forces distributed. The distance between the inclusions is changed by their displacements. Stress distributions around the inclusion and the displacement are shown by numerical calculations.

Solving the Stress Problem for a Finite Cylinder by the Schwarz Method with Hybrid Approximations

An axisymmetric elastic problem is solved by a heterogeneous numerical scheme of the alternating Schwarz method based on the finite- and boundary-element methods. The Schwarz method and the modified scheme are compared. A relaxation parameter is used to accelerate the iterative process

Problem of Contact of Two Half Spaces with a Ring-Shaped Groove in One of Them. Part 2. Numerical Method and Results

A numerical algorithm is proposed for the solution of a singular integral equation of the axisymmetric elastic problem of contact of two half spaces one of which contains a ring-shaped groove. By the methods of orthogonal polynomials and collocations, the singular integral equation is reduced to a system of nonlinear algebraic equations for the geometric parameters of the gap and the coefficients of Fourier expansion of the required function in Chebyshev polynomials of the first kind. A numerical solution of the posed problem is constructed and used for the analysis of contact parameters regarded as functions of the radii of the ring-shaped groove of fixed width.

Crack problem under shear loading in cubic quasicrystal

The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function. Based on the work, the analytic solutions of elastic field of cubic quasicrystal with a penny-shaped crack under the shear loading are found, and the stress intensity factor and strain energy release rate are determined.