Special Analytical Solutions for Use in Debond Stress Analysis

Permanent address: Mathematics Department, Imperial College, London S.W.7. Solutions are presented for the three-dimensional problems ofthe debonding of a rigid rod from an incompressible elasticmatrix. Particular attention is given to debond initiation atkey stress singularities in the perfectly bonded configurationand to the case where the rod base is almost or completely debonded.In each of these cases it is possible by a suitable scalingof the co-ordinates to reduce the problem to one in which athree-dimensional stress field is perturbed by a debond geometrywhich can be viewed as two dimensional. This two-dimensionalproblem is then treated by solving a matrix functional equation.The results are compared with finite element results of an experimentalconfiguration of a rod pull-out test.     

The Numerical Evaluation of Particular Solutions for Poisson's Equation

Three numerical methods are described for the evaluation ofparticular solutions of Poisson's equation. The first methoduses the Newton potential, evaluating it using Gaussian quadraturefollowing a judicious change of integration variables. The secondmethod uses a Fourier sine series to evaluate a particular solution,based on using a Fourier sine series expansion of the inhomogeneousterm of the Poisson equation. Both of these methods assume thatthe inhomogeneous term can be extended smoothly to a suitableregion larger than that of the original domain of the differentialequation. The third method assume the inhomogeneous term isapproximated by a polynomial, and then the Poisson equationwith this new inhomogeneous term is solved exactly for a particularsolution.     

The numerical solution of a non-linear boundary integral equation on smooth surfaces

We study a boundary integral equation method for solving Laplace'sequation u=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with u on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to u. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287.     

On the Discrete Galerkin Method for Fredholm Integral Equations of the Second Kind

In the present paper, we refine some previous results on thediscrete Galerlcin method and the discrete iterated Galerkinmethod for Fredholm integral equations of the second kind. Byconsidering discrete inner products and discrete projectionson the same node points but with different quadrature rules,we are able to treat more appropriately kernels with discontinuousderivatives. In particular, for Green's function kernels weobtain a Nystr?m-type method which has the same order of convergenceas the corresponding Nystr?m method for infinitely smooth kernels.     

Stress Singularities in Viscoelastic Media 2. Plane-Strain Stress Singularities at Corners

The stress singularities that evolve at the corner of a notchedviscoelastic angular plate subject to mode I deformation isdiscussed when prescribed, but arbitrary, displacements aresymmetrically applied to both radial edges of the sector. Thesolution procedure, based on Laplace and Mellin transforms (withtransform parameters p and s, respectively), leads to an eigenvalueproblem in the complex p-plane, which is dependent on Poisson'sratio and characterizes the singular behaviour of the stressfields. Although simple solutions to the transcendental eigenequationare not available, the real-time evolution of the stress concentrationsis obtained by monitoring a particular branch of the eigenvalueequation as it moves in the complex p-plane. A correspondingpath is traced in an appropriately cut strip in the s-plane,in which solutions to the eigenequation are single-valued. Analyticcontinuation in the p-plane thus allows the Laplace and Mellininversions to be performed and the real-time behaviour of theplane-stress components to be expressed as contour integralswithin the strips in the complex s-plane. Cast in this form,the stress components are evaluated numerically when the viscoelasticmaterial is represented as a standard linear solid. Their dependenceon the angular variation within the plate, the applied load,and the effects of the viscoelastic material properties is exhibitedfor a number of situations, and in each case contrasted withshort- and long-time asymptotic curves based on Tauberian theorems.