Solution Of Plane Elasticity Problems Research Articles

Stability analysis of a nanopatterned bimaterial interface

In the article it is shown that the nanopatterned interface of bimaterial is unstable due to the diffusion atom flux along the interface. The main goal of the research is to analyze the conditions of interface stability. The authors developed a model coupling thermodynamics and solid mechanics frameworks. In accordance with the Gurtin—Murdoch theory of surface/interface elasticity, the interphase between two materials is considered as a negligibly thin layer with the elastic properties differing from those of the bulk materials. The growth rate of interface roughness depends on the variation of the chemical potential at the curved interface, which is a function of interface and bulk stresses. The stress distribution along the interface is found from the solution of plane elasticity problem taking into account plane strain conditions. Following this, the linearized evolution equation is derived, which describes the amplitude change of interface perturbation with time.

The Newton-Kantorovich method in analytical solution of plane elasticity problems under superimposed finite strains

The Newton-Kantorovich method in analytical solution of plane elasticity problems under superimposed finite strains

Polynomial wavelets in hybrid‐mixed stress finite element models

AbstractThis paper reports the use of polynomial wavelets as approximation functions in hybrid‐mixed stress finite element models applied to the solution of plane elasticity problems. The stress and displacement fields in the domain and the displacements on the static boundary are independently approximated. The kinematic boundary conditions are locally satisfied. All remaining equations are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of continuum systems, namely the static‐kinematic duality and elastic reciprocity. A set of numerical applications is presented to illustrate the use of the hybrid‐mixed model and to assess its efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

Stress intensity factors for asymmetric branched cracks in plane extension by using crack-tip displacement discontinuity elements

Stress intensity factors are important in the analysis of cracked materials. They are directly related to the fracture propagation and fatigue crack growth criteria. Based on the analytical solution (Crouch, S.L., 1976. Solution of plane elasticity problems by displacement discontinuity method, Int. J. Numer. Methods Eng. 10, pp. 301–343; Crouch, S.L., Starfield, A.M., 1983. Boundary Element Method in Solid Mechanics, with Application in Rock Mechanics and Geological Mechanics, London, Geore Allon and Unwin, Bonton, Sydney) to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid, recently, the crack-tip displacement discontinuity element which can be classified as the left and right crack-tip displacement discontinuity elements are developed by the author Yan, X., (in press. A special crack-tip displacement discontinuity element, Mechanics Research Communications) to model the crack-tip fields to more accurately compute the stress intensity factors of cracks in general plane elasticity. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. To prove further the efficiency of the suggested approach and provide more results of the stress intensity factors, in this study, analysis of an asymmetric branched crack bifurcated from a main crack in plane extension is carried out.

Solution of plane elasticity problems for orthotropic bodies with cracks by the displacement discontinuity method

The force problem of fracture mechanics for orthotropic body is solved by boundary element method. The equations of plane strain state are used. The key point of this paper is obtaining the influence functions for finite segment with constant displacement discontinuities in the main axes of orthotropy. They have been deduced by means of two-dimensional Fourier transform and theory of generalized functions. The straight crack under the action of uniform tension in the infinity is considered by using obtained influence functions. The calculations were carried out for isotropic and two orthotropic materials.

Wavelet Solution of Plane Elasticity Problem in the Upper Half-plane

The plane elasticity problem includes plane strain problem and plane stress problem which are widely applied in mechanics and engineering. In this article, we first reduce the plane elasticity problem in the upper half-plane into natural boundary integral equation and then apply wavelet-Galerkin method to deal with the numerical solution of the natural boundary integral equation. The test and trial functions used are the scaling basis functions of Shannon wavelet. In our fast algorithm, the computational formulae of entries of the stiffness matrix yield simple close-form and only 3 K entries need to be computed for one 4 K ‐ 4 K stiffness matrix.

Solution of plane elasticity problems with Mathematica

The classical complex-variable method of Muskhelishvili for the solution of general plane elasticity problems is revisited. The boundary conditions are satisfied automatically and the related functional equation of Muskhelishvili is approximately solved by selection of the collocation points outside the elastic medium. This ensures the approximate solution of the biharmonic equation of plane elasticity. This approach is completely different from boundary element and finite element methods, since the main attempt is to satisfy the analyticity of the complex potentials (almost equivalently, the biharmonic equation) and not the boundary conditions or both of these. The modern and powerful computer algebra system Mathematica was selected as an appropriate programming language for the related computer procedure, which is also given. Symbolic parameters can also be used in the computing environment offered by Mathematica. Numerical and semi-numerical results in a special application (elliptical region) are also presented and seen to converge rapidly. Both cases of the first and the second fundamental problems are considered.

On the experimental solution of plane elasticity problems by the method of caustics

The optical method of caustics is used for the complete experimental determination of the stress field inside a plane isotropic elastic medium. Two experiments are required for this determination.

Mode I stress intensity factors at corner points in plane elastic media

A well-known method for the solution of plane elasticity problems consists in reducing them to singular integral equations with Cauchy-type kernels. In this paper a numerical technique is presented for the efficient solution of such equations near corner points and the determination of the corresponding stress intensity factors under symmetry conditions so that only Mode I stress intensity factors exist. The peculiarity of the method consists in taking into account all poles of the kernel of a singular integral equation and at the same time using collocation points outside the integration interval. Two applications of the method to V-notched and cracked media are also made.

Solution of elastic crack problems by superposition of finite elements and singular fields

A new finite element method is presented for the solution of plane elasticity problems which contain nonremovable stress singularities. Singular stress field are combined with finite element solutions by a superposition technique; an important feature of the method is that use of the singular fields may be restricted to any specified group of elements which include the singular point. It is shown that good estimates for stress intensity factors are obtained when the method is applied to crack problems.

Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution

AbstractThis paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.

The use of a reflection in the direct solution of plane elasticity problems on multiply-connected domains

The use of a reflection in the direct solution of plane elasticity problems on multiply-connected domains

Numerical Solution of Plane Elasticity Problems

A method is presented for obtaining the stresses in a plane caused by a temperature field, body forces, and boundary forces. The method, based on finite differences, makes efficient use of the digital computer in solving for the irregular boundary shapes often encountered in engineering design. A numerical example shows its application to a notched tension member. (auth)

The method of interface stress adjustment and its uses in the solution of some plane elasticity problems

A method of solution of plane elasticity problems, which utilizes known, closed-form solutions and a finite-difference correction technique, is presented. The method, which simplifies the required computation, is particularly adaptable to problems involving semi-infinite media, singularities, and thermal discontinuities. Some applications are included.