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ASYMPTOTIC ELASTIC STRESS FIELDS AT SINGULAR POINTS

Singular elastic stress fields are generally developed at sharp re-entrant corners and at the end of bonded interfaces between dissimilar elastic materials. This behaviour can present difficulties in both analytical and numerical solution of such problems. For example, excessive mesh refinement might be needed in a finite element solution.

Williams (1952) pioneered a method for determining the strength of the dominant singularity by expressing the local field as an asymptotic expansion. The same method has since been used for a variety of situations leading to singular points, including bonded dissimilar wedges and frictionless or frictional contact between bodies with sharp corners.

Information about the strength of the singularity can be used in analytical solutions to choose an appropriate representation for the fields (for example in the choice of quadrature to use in an integral equation formulation of the problem). It can also be used in numerical solutions to suggest the most appropriate form of graded mesh refinement into the corner, or (better) to develop special corner elements with the analytically determined form. However, asymptotic analysis is seldom the primary purpose of such research and it is tempting just to use a large number of elements in the corner and hope for the best.

We have recently developed a Matlab tool for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity, using Williams' method. The basic mathematics for this procedure is given in Section 11.2 of J.R.Barber, Elasticity, Kluwer, Dordrecht 2nd edn. (2002), 410pp. However, it is not necessary to have any detailed knowledge of the method in order to use the tool.

A more detailed description of the procedure, including detailed instructions for using the analytical tool has been published in the Journal of Strain Analysis for Engineering Design and can be downloaded here.

The user is prompted to input the local geometry of the system, the material properties and the boundary conditions (and interface conditions in the case of composite bodies or problems involving contact between two or more bodies). The tool then computes the dominant eigenvalue and provides as output the equations defining the singular stress and displacement fields and contour plots of these fields. No knowledge of the asymptotic analysis procedure is required of the user.

The tool is written in the software code MATLAB v7.0 with the MATLAB GUI development environment (GUIDE) v2.5 and the MATLAB Symbolic Toolbox v3.1. It provides a graphic interface in which users can define their problem, determine the order of the corresponding singularity and generate the distribution of stress and displacement. Final results are provided in both text and graphic format.

To download the source code, click on this link and unzip the downloaded file. If and only if you have difficulty opening this link, try this one . After downloading and unzipping the file, open MATLAB and start the program with the command `ws'.

Two example programs are included: `williams.wat' and `bogy.wat', which solve the problems of the single wedge and the bi-material wedge respectively.

Please report any problems with the software or any suggestions for additional features or improvements to Donghee Lee and J.R.Barber

 

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