Axisymmetric Elasticity Problems Research Articles

Traction‐associated peridynamic model and non‐uniform discretization simulation of plane axisymmetric problems

AbstractUsing the recently developed traction‐associated peridynamic motion equation (TPME), we investigate the plane axisymmetric problems of elastic deformation. The polar coordinate form of TPME is derived for a plane axisymmetric problem. The transfer function matching with prototype microelastic (PM) model is determined by the inverse method in a one‐dimensional case, and then it is directly extended to plane axisymmetric elasticity problems and plane problems. We promote a strategy to deal with the horizon in the non‐uniform discretization scheme. Based on this strategy, the non‐uniform discretization scheme is used to solve the elastic deformation of a hollow circular plate subjected to uniform inner and outer pressures. Total process from elastic deformation to failure of the hollow circular plate under tension is simulated completely. The results show that TPME does not only need volume and surface corrections, but also can deal with the complex traction boundary conditions conveniently.

Elastic and thermoelastic response of multilayer inhomogeneous hollow cylinders

An approach is presented for solving plane axisymmetric problems of elasticity and thermoelasticity for multilayer hollow cylinders with per-layer properties having arbitrary distribution profiles within the radial coordinate. The approach is based on the representation of material profiles within the entire assembly in the form of stepwise-variable functions with the implication of the direct integration method toward the compatibility equation in terms of strains. The approach avoids using the generalized derivative apparatus. The original problems are reduced to a governing integral equation with an integral condition implying a solution through explicit dependence on the force and thermal loadings.

A New General Solution of Axisymmetric Elastic Space Problem in Pavement Engineering

The general solution of the axial symmetry elastic space problem in cement concrete pavement is an elementary question. Combining the Love method and the Southwell method, the Southwell operator was used in the variable selection process, the displacement function was introduced to express the displacement component by the Love method, the expression of stress indicated by the displacement function was obtained by combining the displacement components, geometric equations, and physical equations, then the stress was substituted into the equilibrium equation, the biharmonic equation of displacement function was obtained by mathematical operation, and finally, a new general solution of the axisymmetric elastic space problem in cement concrete pavement was obtained. According to the proposed general solution, mechanical calculation for a cement concrete pavement slab on Winkler foundation was carried out, the displacement curve at the top surface of the slab and the stress curve at the top, middle, and bottom surface of the slab were derived, and the results showed that the method is feasible in the calculation of cement concrete pavement. The general solution provided a new method for solving the axisymmetric elastic space problem of cement concrete pavement.

Integration of the equations of plane axisymmetric problems of elasticity and thermoelasticity for layered solid cylinders

Integration of the equations of plane axisymmetric problems of elasticity and thermoelasticity for layered solid cylinders

Stress analysis of composites with interfacial bridged cracks and damaged zones

Stress analysis of composite structures with bridged cracks and damaged zones is performed with the direct boundary integral equations method in axisymmetric formulation. It is assumed that: a) the crack faces interact in some zones behind of the cracks edges (bridged zones); b) these zones width can be comparable to the whole crack size; c) adjacent parts of the structure can be connected by mechanical ligaments (including interfaces ahead of cracks fronts) and the relative displacements of these parts can occur under the loading (these zones consider as damaged zones). The boundary element software to solve axisymmetric problems of elasticity with interfacial bridged cracks and damaged zones was developed. The results for different properties of composite materials and bonds are presented. Applications the bridged crack approach to modeling of cracks self-healing is remarked.

A generalized solution to the combo-crack problem—I. Pressure load on crack surface

The axisymmetric elasticity problem of crack combo containing an externally circular crack (ECC) and a coplanar concentric penny-shaped crack (PSC) is mathematically equivalent to the annular contact problem. This problem has been attempted by using Love's strain potential approach, which eventually comes down to solving a pair of simultaneous Fredholm integral equations. Finding the closed-form solutions to the integral equations is difficult, if not impossible. Approximate solutions have been proposed in power series representations, which suffer from two major deficiencies. First, the solutions apply only to a special loading case in which uniform pressure is applied to the whole surface of the interior PSC. Secondly, the accuracy of the solution becomes unsatisfactory when the interior PSC tip is close to the ECC tip. To address these issues, in this paper we revisit this problem by considering a more general loading case in which uniform pressure is applied to a circular region of any size at the center of the PSC's surface. To overcome the lower accuracy caused by power series with limited terms, we numerically solve the pair of simultaneous Fredholm integral equations based on the Gauss-Lobatto quadrature. The high accuracy of our solution in the whole size spectra of the PSC and ECC is verified by finite element simulations. Our paper provides a generalized and more accurate solution to the annular contact problem or the combo crack problem, which deserves to be included in the updated library of the solutions to basic crack problems.

The use of the variational method of homogeneous solutions in an axisymmetric elasticity problem for a finite cylinder under its weight

The use of the variational method of homogeneous solutions in an axisymmetric elasticity problem for a finite cylinder under its weight

Hybrid Trefftz finite element method for axisymmetric elasticity problems under torsion

An eight-node axisymmetric element AX88T is proposed for predicting the torsional response of axisymmetric elasticity based on the hybrid fundamental solution based finite element method (HFS-FEM). In this approach, two displacement fields are independently assumed in the domain and on the boundary of the element, respectively. The intra-element displacement field is expressed in terms of fundamental solutions satisfying the governing equation of the problem while a conforming frame displacement field is related to the nodal degrees of freedom by standard one-dimensional interpolation functions. By applying the Gauss divergence theorem to the hybrid functional, the resultant finite element formulation completely avoids the need for domain integration, which renders the AX88T element insensitive to distortion. The accuracy and efficiency of HFS-FEM are demonstrated through three numerical examples which includes thick-walled cylinder, flange-shaped cylinder and stepped cylindrical shaft.

Axisymmetric Elastic Problem of a Penny-shaped Crack in a Multi-layer Composite

Axisymmetric Elastic Problem of a Penny-shaped Crack in a Multi-layer Composite

On the Kelvin, Boussinesq, and Mindlin problems

Three fundamental three-dimensional axisymmetric problems of elasticity are revisited: the Kelvin problem of a concentrated force in an infinite space, the Boussinesq problem of a vertical concentrated force at the boundary of a half-space, and the Mindlin problem of a vertical concentrated force in the interior of a half-space. New elements of the derivation of the solution to each of these problems are included to make the presentation more appealing for the coverage of the topic in graduate courses of solid mechanics. Two approaches are employed and compared, one based on the Galerkin method and Love’s potential function, and the other based on the Papkovich–Neuber displacement representation and Boussinesq’s potential functions. A historical perspective and a referral to more recent contributions in the field are also given.

Correspondence principle in plane and axisymmetric mixed boundary-value problems of elasticity

This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.

Solutions of Axisymmetric Problems of Elasticity and Thermoelasticity for an Inhomogeneous Space and a Half Space

We developed a technique for the construction of solutions of axisymmetric problems of elasticity and thermoelasticity in stresses for a space and a half space whose elastic properties are arbitrary functions of the coordinate z. By using the direct integration method and the Hankel integral transformation, the problems are reduced to governing integral equations accompanied by a local boundary condition and an integral condition in the case of the half space. The solutions of the deduced equations are constructed in the explicit form by using the resolvent-kernel method.

A hybrid fundamental-solution-based 8-node element for axisymmetric elasticity problems

A novel hybrid finite element approach using fundamental solutions in conjunction with the boundary integration is proposed for the analysis of axisymmetric elasticity problems. In the formulation two displacement fields are independently assumed within the element domain and at the element boundary, which combines the advantages of conventional finite and boundary element methods. The resulting system of algebraic equations is symmetric and positive definite, which reduces the spatial dimension of the problem by one. It offers great flexibility in generating the mesh with arbitrary higher-order elements. Firstly, the proposed approach is validated by the analytical solution of a thick-walled hollow cylinder. A good agreement is observed. Meantime, the optimal number and location of source points for fundamental solutions as well as sensitivity to mesh distortion are studied. Finally, two more numerical examples including a flange and a wheel are provided. Compared to ABAQUS, the proposed approach exhibits higher efficiency, better convergence rate and strong capability to capture geometric singularities.

Варіаційний метод однорідних розв’язків у осесиметричній задачі теорії пружності для скінченного циліндра

Варіаційний метод однорідних розв’язків у осесиметричній задачі теорії пружності для скінченного циліндра

Cosserat Spectrum of an Axisymmetric Elasticity Problem for a Finite-Length Solid Cylinder

ABSTRACTAn algorithm for the computation and analysis of the Cosserat spectrum for an axisymmetric elasticity boundary-value problem in a finite-length solid cylinder with boundary conditions in terms of stresses is proposed. By making use of the cross-wise superposition method, the spectral problem is reduced to systems of linear algebraic equations. A solution method for the mentioned systems is presented and the asymptotic behavior of the Cosserat eigenvalues is established. On this basis, the key features of the Cosserat spectrum for the mentioned problem are analyzed with special attention given to the effect of the cylinder aspect ratio.

BESSEL FUNCTIONS FOR AXISYMMETRIC ELASTICITY PROBLEMS OF THE ELASTIC HALF SPACE SOIL: A POTENTIAL FUNCTION METHOD

Elasticity problems are formulated using displacement methods or stress methods. In this paper a displacement formulation of axisymmetric elasticity problem is presented. The formulation uses the Boussinesq– Papkovich – Neuber potential function. The problem is then solved by assuming Boussinesq – Papkovich - Neuber potential functions in the form of Bessel functions of order zero and of the first kind. The potential functions are then made to satisfy the governing field equations and the associated boundary conditions for the particular problem of a point load at the origin of the semi-infinite linear elastic isotropic soil mass. The unknown parameters of the function are thus determined and used to find the stresses, strains and displacement fields in the loaded soil. The results obtained were identical with the results obtained by Boussinesq. http://dx.doi.org/10.4314/njt.v36i3.16

BESSEL FUNCTIONS FOR AXISYMMETRIC ELASTICITY PROBLEMS OF THE ELASTIC HALF SPACE SOIL: A POTENTIAL FUNCTION METHOD

Elasticity problems are formulated using displacement methods or stress methods. In this paper a displacement formulation of axisymmetric elasticity problem is presented. The formulation uses the Boussinesq– Papkovich – Neuber potential function. The problem is then solved by assuming Boussinesq – Papkovich - Neuber potential functions in the form of Bessel functions of order zero and of the first kind. The potential functions are then made to satisfy the governing field equations and the associated boundary conditions for the particular problem of a point load at the origin of the semi-infinite linear elastic isotropic soil mass. The unknown parameters of the function are thus determined and used to find the stresses, strains and displacement fields in the loaded soil. The results obtained were identical with the results obtained by Boussinesq. http://dx.doi.org/10.4314/njt.v36i3.16

Inverse Solution for Bolt Preload Using Surface Deformation

In this paper, an inverse biharmonic axisymmetric elasticity problem is solved by invoking measured out-of-plane surface deformation values at discrete locations around a preloaded bolt head, in order to calculate the underhead contact stress and joint clamp load that would have caused that out-of-plane surface deformation. Solution of this type of inverse problem promises to improve the automation process of bolted joint system assembly, especially in critical and safety-related applications. For example, a real-time optically measured joint surface deformation can be utilized for automating process control of bolted joint assembly in a reliable fashion. This would be a significant reliability improvement as compared to the commonly used method in mass production using torque-only control method in which there is wide scatter in the torque–tension correlation due to the normal scatter in frictional variables. Finite element analysis (FEA) method is used to validate the inverse problem solution provided in this paper.

Axisymmetric Stresses in an Elastic Radially Inhomogeneous Cylinder Under Length-Varying Loadings

In this paper, we present an analytical solution to the axisymmetric elasticity problem for an inhomogeneous solid cylinder subjected to external force loadings, which vary within the axial coordinate. The material properties of the cylinder are assumed to be arbitrary functions of the radial coordinate. By making use of the direct integration method, the problem is reduced to coupled integral equations for the shearing stress and the total stress (given by the superposition of the normal ones). By making use of the resolvent-kernel solution, the latter equations were uncoupled and then solved in a closed analytical form. On this basis, the effect of variable material moduli in the stress distribution has been examined with special attention given to the negative Poisson's ratio.

Effect of the layer inhomogeneity on the distribution of stresses and displacements in an elastic multilayer cylinder

We present an analytical approach for constructing a solution to the plane axisymmetric elasticity and thermoelasticity problems for a multilayer solid cylinder with arbitrarily variable material properties within each layer. The problems were attempted by means of the direct integration method along with the generalized functions technique. By this means, an explicit-form solution is obtained which allows for the adequate analysis of stresses and displacements in multilayer nonhomogeneous cylinders. Special attention is given to the interlayer stress distributions for the cases of discontinuous, continuous, and smooth variation of the material properties.